Applications of Linear Equations

Linear EquationsPower Equations

Applications of Linear Equations

Linear equations are essential tools for modeling and solving problems that involve a constant rate of change. In this section, we explore five practical applications, demonstrating how they help us make calculations, predictions, and understand various real-world situations.

Cost and Revenue Analysis

Businesses often use linear equations to analyze the relationship between costs (expenses) and revenue (income) as a function of the number of units produced or sold. Understanding these relationships is crucial for determining profitability and making informed business decisions.

A small bakery sells cupcakes. The fixed costs (rent, utilities) are 1.50. They sell each cupcake for CxRxCYou can't use 'macro parameter character #' in math mode, is the sum of fixed costs and variable costs. The variable cost is \$1.50 per cupcake, therefore <mjx-container aria-label="C = 1.50x + 200" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="16.213ex" height="1.781ex" role="img" focusable="false" viewBox="0 -705 7166 787" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D436" xlink:href="#MJX-TEX-I-1D436"></use></g><g data-mml-node="mo" transform="translate(1037.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2093.6,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(778,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1278,0)"></use></g><g data-mml-node="mi" transform="translate(3871.6,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(4665.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(5666,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1000,0)"></use></g></g></g></svg></mjx-container> 2. The revenue,RYou can't use 'macro parameter character #' in math mode, is the amount earned from selling cupcakes. Since each cupcake sells for \$3.50: <mjx-container aria-label="R = 3.50x" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="10.051ex" height="1.731ex" role="img" focusable="false" viewBox="0 -683 4442.6 765" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D445" xlink:href="#MJX-TEX-I-1D445"></use></g><g data-mml-node="mo" transform="translate(1036.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(2092.6,0)"><use data-c="33" xlink:href="#MJX-TEX-N-33"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(778,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1278,0)"></use></g><g data-mml-node="mi" transform="translate(3870.6,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g></g></g></svg></mjx-container> 3. The break-even point is when cost equals revenue, i.e.,C = RYou can't use 'macro parameter character #' in math mode. Thus we solve: <mjx-container aria-label="\begin{align*} 1.50x + 200 &= 3.50x \\ 200 &= 2.00x \\ x &= 100 \end{align*}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -3.507ex;" xmlns="http://www.w3.org/2000/svg" width="19.81ex" height="8.145ex" role="img" focusable="false" viewBox="0 -2050 8756 3600" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtable"><g data-mml-node="mtr" transform="translate(0,1300)"><g data-mml-node="mtd"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(778,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1278,0)"></use></g><g data-mml-node="mi" transform="translate(1778,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(2572.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(3572.4,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1000,0)"></use></g></g><g data-mml-node="mtd" transform="translate(5072.4,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1333.6,0)"><use data-c="33" xlink:href="#MJX-TEX-N-33"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(778,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1278,0)"></use></g><g data-mml-node="mi" transform="translate(3111.6,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,0)"><g data-mml-node="mtd" transform="translate(3572.4,0)"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1000,0)"></use></g></g><g data-mml-node="mtd" transform="translate(5072.4,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1333.6,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(778,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1278,0)"></use></g><g data-mml-node="mi" transform="translate(3111.6,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-1300)"><g data-mml-node="mtd" transform="translate(4500.4,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g></g><g data-mml-node="mtd" transform="translate(5072.4,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1333.6,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1000,0)"></use></g></g></g></g></g></g></svg></mjx-container> Therefore, the bakery breaks even when they sell 100 cupcakes. </details></div><h2>Distance, Speed, and Time Problems</h2> <p>Linear equations can be used to solve problems involving motion at a constant speed. The fundamental relationship between average speed, distance, and time is given by: </p> <mjx-container aria-label="\text{ speed} =\frac{\text{distance traveled}}{\text{time}}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -1.577ex;" xmlns="http://www.w3.org/2000/svg" width="26.605ex" height="4.676ex" role="img" focusable="false" viewBox="0 -1370 11759.6 2067" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtext"><use data-c="A0" xlink:href="#MJX-TEX-N-A0"></use><use data-c="73" xlink:href="#MJX-TEX-N-73" transform="translate(250,0)"></use><use data-c="70" xlink:href="#MJX-TEX-N-70" transform="translate(644,0)"></use><use data-c="65" xlink:href="#MJX-TEX-N-65" transform="translate(1200,0)"></use><use data-c="65" xlink:href="#MJX-TEX-N-65" transform="translate(1644,0)"></use><use data-c="64" xlink:href="#MJX-TEX-N-64" transform="translate(2088,0)"></use></g><g data-mml-node="mo" transform="translate(2921.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(3977.6,0)"><g data-mml-node="mtext" transform="translate(220,676)"><use data-c="64" xlink:href="#MJX-TEX-N-64"></use><use data-c="69" xlink:href="#MJX-TEX-N-69" transform="translate(556,0)"></use><use data-c="73" xlink:href="#MJX-TEX-N-73" transform="translate(834,0)"></use><use data-c="74" xlink:href="#MJX-TEX-N-74" transform="translate(1228,0)"></use><use data-c="61" xlink:href="#MJX-TEX-N-61" transform="translate(1617,0)"></use><use data-c="6E" xlink:href="#MJX-TEX-N-6E" transform="translate(2117,0)"></use><use data-c="63" xlink:href="#MJX-TEX-N-63" transform="translate(2673,0)"></use><use data-c="65" xlink:href="#MJX-TEX-N-65" transform="translate(3117,0)"></use><use data-c="20" xlink:href="#MJX-TEX-N-20" transform="translate(3561,0)"></use><use data-c="74" xlink:href="#MJX-TEX-N-74" transform="translate(3811,0)"></use><use data-c="72" xlink:href="#MJX-TEX-N-72" transform="translate(4200,0)"></use><use data-c="61" xlink:href="#MJX-TEX-N-61" transform="translate(4592,0)"></use><use data-c="76" xlink:href="#MJX-TEX-N-76" transform="translate(5092,0)"></use><use data-c="65" xlink:href="#MJX-TEX-N-65" transform="translate(5620,0)"></use><use data-c="6C" xlink:href="#MJX-TEX-N-6C" transform="translate(6064,0)"></use><use data-c="65" xlink:href="#MJX-TEX-N-65" transform="translate(6342,0)"></use><use data-c="64" xlink:href="#MJX-TEX-N-64" transform="translate(6786,0)"></use></g><g data-mml-node="mtext" transform="translate(2919,-686)"><use data-c="74" xlink:href="#MJX-TEX-N-74"></use><use data-c="69" xlink:href="#MJX-TEX-N-69" transform="translate(389,0)"></use><use data-c="6D" xlink:href="#MJX-TEX-N-6D" transform="translate(667,0)"></use><use data-c="65" xlink:href="#MJX-TEX-N-65" transform="translate(1500,0)"></use></g><rect width="7542" height="60" x="120" y="220"></rect></g></g></g></svg></mjx-container><p> orr=d/tdrtt70t90tYou can't use 'macro parameter character #' in math mode. The sum of the distances will equal the total distance between the trains which is 500 miles. Therefore: <mjx-container aria-label="\begin{align*} 70t + 90t &= 500 \\ 160t &= 500 \\ t &= \frac{500}{160} = 3.125 \end{align*}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -4.695ex;" xmlns="http://www.w3.org/2000/svg" width="24.501ex" height="10.52ex" role="img" focusable="false" viewBox="0 -2575 10829.6 4650" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtable"><g data-mml-node="mtr" transform="translate(0,1825)"><g data-mml-node="mtd"><g data-mml-node="mn"><use data-c="37" xlink:href="#MJX-TEX-N-37"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mi" transform="translate(1000,0)"><use data-c="1D461" xlink:href="#MJX-TEX-I-1D461"></use></g><g data-mml-node="mo" transform="translate(1583.2,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(2583.4,0)"><use data-c="39" xlink:href="#MJX-TEX-N-39"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mi" transform="translate(3583.4,0)"><use data-c="1D461" xlink:href="#MJX-TEX-I-1D461"></use></g></g><g data-mml-node="mtd" transform="translate(3944.4,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1333.6,0)"><use data-c="35" xlink:href="#MJX-TEX-N-35"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1000,0)"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,525)"><g data-mml-node="mtd" transform="translate(2083.4,0)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="36" xlink:href="#MJX-TEX-N-36" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mi" transform="translate(1500,0)"><use data-c="1D461" xlink:href="#MJX-TEX-I-1D461"></use></g></g><g data-mml-node="mtd" transform="translate(3944.4,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1333.6,0)"><use data-c="35" xlink:href="#MJX-TEX-N-35"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1000,0)"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-1367)"><g data-mml-node="mtd" transform="translate(3583.4,0)"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-TEX-I-1D461"></use></g></g><g data-mml-node="mtd" transform="translate(3944.4,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(1333.6,0)"><g data-mml-node="mn" transform="translate(220,676)"><use data-c="35" xlink:href="#MJX-TEX-N-35"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mn" transform="translate(220,-686)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="36" xlink:href="#MJX-TEX-N-36" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1000,0)"></use></g><rect width="1700" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(3551.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(4607.1,0)"><use data-c="33" xlink:href="#MJX-TEX-N-33"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use><use data-c="31" xlink:href="#MJX-TEX-N-31" transform="translate(778,0)"></use><use data-c="32" xlink:href="#MJX-TEX-N-32" transform="translate(1278,0)"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(1778,0)"></use></g></g></g></g></g></g></svg></mjx-container> It will take 3.125 hours (or 3 hours and 7.5 minutes) for the trains to be 500 miles apart. </details></div><div class="example"><p>The distance between two cities is 200 kilometers. A car goes from city A to city B with an average speed of 100 kilometers per hour and immediately comes back from city B to city A with an average speed of 80 kilometers per hour. What is the average speed of this car for the entire trip?</p> <details> <summary>Solution</summary> First, note that the average speed is not equal to the average of the speeds. The average speed for the entire trip is equal to the total distance divided by the total time. Lett_1t_2You can't use 'macro parameter character #' in math modethe time to come back from B to A. The time is equal to distance divided by the average speed so <mjx-container aria-label="t_1 = \frac{200 \text{ km}}{100 \text{ km/h}} = 2 \text{ h}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -2.172ex;" xmlns="http://www.w3.org/2000/svg" width="21.216ex" height="5.271ex" role="img" focusable="false" viewBox="0 -1370 9377.7 2330" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-TEX-I-1D461"></use></g><g data-mml-node="mn" transform="translate(394,-150) scale(0.707)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g></g><g data-mml-node="mo" transform="translate(1075.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(2131.1,0)"><g data-mml-node="mrow" transform="translate(748,676)"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mtext" transform="translate(1500,0)"><use data-c="A0" xlink:href="#MJX-TEX-N-A0"></use><use data-c="6B" xlink:href="#MJX-TEX-N-6B" transform="translate(250,0)"></use><use data-c="6D" xlink:href="#MJX-TEX-N-6D" transform="translate(778,0)"></use></g></g><g data-mml-node="mrow" transform="translate(220,-710)"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mtext" transform="translate(1500,0)"><use data-c="A0" xlink:href="#MJX-TEX-N-A0"></use><use data-c="6B" xlink:href="#MJX-TEX-N-6B" transform="translate(250,0)"></use><use data-c="6D" xlink:href="#MJX-TEX-N-6D" transform="translate(778,0)"></use><use data-c="2F" xlink:href="#MJX-TEX-N-2F" transform="translate(1611,0)"></use><use data-c="68" xlink:href="#MJX-TEX-N-68" transform="translate(2111,0)"></use></g></g><rect width="4367" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(7015.9,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(8071.7,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g><g data-mml-node="mtext" transform="translate(8571.7,0)"><use data-c="A0" xlink:href="#MJX-TEX-N-A0"></use><use data-c="68" xlink:href="#MJX-TEX-N-68" transform="translate(250,0)"></use></g></g></g></svg></mjx-container> <mjx-container aria-label="t_2 = \frac{200 \text{ km}}{80 \text{ km/h}} = 2.5 \text{ h}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -2.172ex;" xmlns="http://www.w3.org/2000/svg" width="21.845ex" height="5.271ex" role="img" focusable="false" viewBox="0 -1370 9655.7 2330" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="msub"><g data-mml-node="mi"><use data-c="1D461" xlink:href="#MJX-TEX-I-1D461"></use></g><g data-mml-node="mn" transform="translate(394,-150) scale(0.707)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use></g></g><g data-mml-node="mo" transform="translate(1075.3,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mfrac" transform="translate(2131.1,0)"><g data-mml-node="mrow" transform="translate(498,676)"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mtext" transform="translate(1500,0)"><use data-c="A0" xlink:href="#MJX-TEX-N-A0"></use><use data-c="6B" xlink:href="#MJX-TEX-N-6B" transform="translate(250,0)"></use><use data-c="6D" xlink:href="#MJX-TEX-N-6D" transform="translate(778,0)"></use></g></g><g data-mml-node="mrow" transform="translate(220,-710)"><g data-mml-node="mn"><use data-c="38" xlink:href="#MJX-TEX-N-38"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mtext" transform="translate(1000,0)"><use data-c="A0" xlink:href="#MJX-TEX-N-A0"></use><use data-c="6B" xlink:href="#MJX-TEX-N-6B" transform="translate(250,0)"></use><use data-c="6D" xlink:href="#MJX-TEX-N-6D" transform="translate(778,0)"></use><use data-c="2F" xlink:href="#MJX-TEX-N-2F" transform="translate(1611,0)"></use><use data-c="68" xlink:href="#MJX-TEX-N-68" transform="translate(2111,0)"></use></g></g><rect width="3867" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(6515.9,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(7571.7,0)"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(778,0)"></use></g><g data-mml-node="mtext" transform="translate(8849.7,0)"><use data-c="A0" xlink:href="#MJX-TEX-N-A0"></use><use data-c="68" xlink:href="#MJX-TEX-N-68" transform="translate(250,0)"></use></g></g></g></svg></mjx-container> The total distance is200 + 200 = 4002 + 2.5 = 4.5You can't use 'macro parameter character #' in math modehours. So the average speed for the whole trip is <mjx-container aria-label="\frac{400 \text{ km}}{4.5 \text{ h}} \approx 88.89 \text{ km/h}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -1.602ex;" xmlns="http://www.w3.org/2000/svg" width="22.239ex" height="4.701ex" role="img" focusable="false" viewBox="0 -1370 9829.6 2078" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mfrac"><g data-mml-node="mrow" transform="translate(220,676)"><g data-mml-node="mn"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1000,0)"></use></g><g data-mml-node="mtext" transform="translate(1500,0)"><use data-c="A0" xlink:href="#MJX-TEX-N-A0"></use><use data-c="6B" xlink:href="#MJX-TEX-N-6B" transform="translate(250,0)"></use><use data-c="6D" xlink:href="#MJX-TEX-N-6D" transform="translate(778,0)"></use></g></g><g data-mml-node="mrow" transform="translate(733.5,-686)"><g data-mml-node="mn"><use data-c="34" xlink:href="#MJX-TEX-N-34"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(778,0)"></use></g><g data-mml-node="mtext" transform="translate(1278,0)"><use data-c="A0" xlink:href="#MJX-TEX-N-A0"></use><use data-c="68" xlink:href="#MJX-TEX-N-68" transform="translate(250,0)"></use></g></g><rect width="3311" height="60" x="120" y="220"></rect></g><g data-mml-node="mo" transform="translate(3828.8,0)"><use data-c="2248" xlink:href="#MJX-TEX-N-2248"></use></g><g data-mml-node="mn" transform="translate(4884.6,0)"><use data-c="38" xlink:href="#MJX-TEX-N-38"></use><use data-c="38" xlink:href="#MJX-TEX-N-38" transform="translate(500,0)"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(1000,0)"></use><use data-c="38" xlink:href="#MJX-TEX-N-38" transform="translate(1278,0)"></use><use data-c="39" xlink:href="#MJX-TEX-N-39" transform="translate(1778,0)"></use></g><g data-mml-node="mtext" transform="translate(7162.6,0)"><use data-c="A0" xlink:href="#MJX-TEX-N-A0"></use><use data-c="6B" xlink:href="#MJX-TEX-N-6B" transform="translate(250,0)"></use><use data-c="6D" xlink:href="#MJX-TEX-N-6D" transform="translate(778,0)"></use><use data-c="2F" xlink:href="#MJX-TEX-N-2F" transform="translate(1611,0)"></use><use data-c="68" xlink:href="#MJX-TEX-N-68" transform="translate(2111,0)"></use></g></g></g></svg></mjx-container> </details></div><h2>Mixture Problems</h2> <p>Mixture problems involve combining different amounts of a substance to achieve a desired result. These problems can often be solved using a single linear equation if the total mixture amount is known.</p> <div class="example"><p>A chemist has 10 liters of a 25% saline solution. They want to dilute it to a 10% solution by adding pure water. How much pure water (which has a 0</p> <details> <summary>Solution</summary> Letx0.25 \times 10 = 2.510 + x10%0.10(10+x)You can't use 'macro parameter character #' in math mode. Since only water is added, the amount of salt remains constant. Therefore: <mjx-container aria-label="\begin{align*} 2.5 &= 0.10(10+x) \\ 2.5 &= 1 + 0.10x \\ 1.5 &= 0.10x \\ x &= 15 \end{align*}" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -4.977ex;" xmlns="http://www.w3.org/2000/svg" width="18.014ex" height="11.086ex" role="img" focusable="false" viewBox="0 -2700 7962 4900" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mtable"><g data-mml-node="mtr" transform="translate(0,1950)"><g data-mml-node="mtd"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(778,0)"></use></g></g><g data-mml-node="mtd" transform="translate(1278,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1333.6,0)"><use data-c="30" xlink:href="#MJX-TEX-N-30"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use><use data-c="31" xlink:href="#MJX-TEX-N-31" transform="translate(778,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1278,0)"></use></g><g data-mml-node="mo" transform="translate(3111.6,0)"><use data-c="28" xlink:href="#MJX-TEX-N-28"></use></g><g data-mml-node="mn" transform="translate(3500.6,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(500,0)"></use></g><g data-mml-node="mo" transform="translate(4722.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mi" transform="translate(5723,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g><g data-mml-node="mo" transform="translate(6295,0)"><use data-c="29" xlink:href="#MJX-TEX-N-29"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,650)"><g data-mml-node="mtd"><g data-mml-node="mn"><use data-c="32" xlink:href="#MJX-TEX-N-32"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(778,0)"></use></g></g><g data-mml-node="mtd" transform="translate(1278,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1333.6,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use></g><g data-mml-node="mo" transform="translate(2055.8,0)"><use data-c="2B" xlink:href="#MJX-TEX-N-2B"></use></g><g data-mml-node="mn" transform="translate(3056,0)"><use data-c="30" xlink:href="#MJX-TEX-N-30"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use><use data-c="31" xlink:href="#MJX-TEX-N-31" transform="translate(778,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1278,0)"></use></g><g data-mml-node="mi" transform="translate(4834,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-650)"><g data-mml-node="mtd"><g data-mml-node="mn"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(778,0)"></use></g></g><g data-mml-node="mtd" transform="translate(1278,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1333.6,0)"><use data-c="30" xlink:href="#MJX-TEX-N-30"></use><use data-c="2E" xlink:href="#MJX-TEX-N-2E" transform="translate(500,0)"></use><use data-c="31" xlink:href="#MJX-TEX-N-31" transform="translate(778,0)"></use><use data-c="30" xlink:href="#MJX-TEX-N-30" transform="translate(1278,0)"></use></g><g data-mml-node="mi" transform="translate(3111.6,0)"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g></g></g><g data-mml-node="mtr" transform="translate(0,-1950)"><g data-mml-node="mtd" transform="translate(706,0)"><g data-mml-node="mi"><use data-c="1D465" xlink:href="#MJX-TEX-I-1D465"></use></g></g><g data-mml-node="mtd" transform="translate(1278,0)"><g data-mml-node="mi"></g><g data-mml-node="mo" transform="translate(277.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mn" transform="translate(1333.6,0)"><use data-c="31" xlink:href="#MJX-TEX-N-31"></use><use data-c="35" xlink:href="#MJX-TEX-N-35" transform="translate(500,0)"></use></g></g></g></g></g></g></svg></mjx-container> The chemist needs to add 15 liters of water to the solution. </details></div><h2>Simple Interest Calculations</h2> <p>Simple interest is a method of calculating interest where the interest is earned only on the principal amount. The formula for simple interest,IYou can't use 'macro parameter character #' in math mode, is given by: </p> <mjx-container aria-label="I = Prt" class="MathJax" jax="SVG" display="true"><svg style="vertical-align: -0.186ex;" xmlns="http://www.w3.org/2000/svg" width="7.694ex" height="1.731ex" role="img" focusable="false" viewBox="0 -683 3400.6 765" xmlns:xlink="http://www.w3.org/1999/xlink"><g stroke="currentColor" fill="currentColor" stroke-width="0" transform="scale(1,-1)"><g data-mml-node="math"><g data-mml-node="mi"><use data-c="1D43C" xlink:href="#MJX-TEX-I-1D43C"></use></g><g data-mml-node="mo" transform="translate(781.8,0)"><use data-c="3D" xlink:href="#MJX-TEX-N-3D"></use></g><g data-mml-node="mi" transform="translate(1837.6,0)"><use data-c="1D443" xlink:href="#MJX-TEX-I-1D443"></use></g><g data-mml-node="mi" transform="translate(2588.6,0)"><use data-c="1D45F" xlink:href="#MJX-TEX-I-1D45F"></use></g><g data-mml-node="mi" transform="translate(3039.6,0)"><use data-c="1D461" xlink:href="#MJX-TEX-I-1D461"></use></g></g></g></svg></mjx-container><p> wherePrt1000 in a savings account that earns 4% simple annual interest. How much interest will she earn after 3 years?

Solution The simple interest, , is calculated as: where is the principal, is the annual rate, and is the time in years. Plugging in the given values: Sarah will earn \$120 in interest after 3 years.

Linear Depreciation

Linear depreciation is a method used to estimate the loss in value of an asset over time. In a linear depreciation model, the asset loses value at a constant rate over its lifespan. It is a simplification of real-world depreciation scenarios, but it's useful for modeling purposes.

A company purchases a machine for 1,000 after 9 years. Using a linear depreciation model, determine the value of the machine after 5 years.

Solution First, we find the annual depreciation. The machine depreciates by over 9 years, for an annual depreciation of . Let be the value of the machine after years. We can model this with a linear equation: After 5 years, the value is: The value of the machine after 5 years is \$5000.

Temperature Conversion

Linear equations are used to convert temperatures between different scales. For example, the relationship between Celsius () and Fahrenheit () is linear.

The relationship between Celsius and Fahrenheit is given by . If the temperature is 77 degrees Farenheit, what is the temperature in Celsius?

Solution We substitute into the formula: Therefore, 25 degrees Celsius is equal to 77 degrees Fahrenheit.

These examples illustrate just a few of the many ways linear equations can be used to solve real-world problems. Their power lies in their simplicity and ability to model situations involving a constant rate of change. Understanding how to construct and solve linear equations is a fundamental skill in mathematics and is valuable in various fields of study and work.

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