Models and Applications - Revision history

Lila: /* Example: Setting Up a Linear Equation to Solve a Real-World Application */

2021-11-03T15:06:04Z

Example: Setting Up a Linear Equation to Solve a Real-World Application

Lila at 15:04, 3 November 2021

2021-11-03T15:04:22Z

Lila: /* Example: Setting Up a Linear Equation to Solve a Real-World Application */

2021-11-03T15:03:12Z

Example: Setting Up a Linear Equation to Solve a Real-World Application

Lila at 15:02, 3 November 2021

2021-11-03T15:02:19Z

Lila at 15:00, 3 November 2021

2021-11-03T15:00:23Z

Lila at 14:59, 3 November 2021

2021-11-03T14:59:56Z

Lila at 14:57, 3 November 2021

2021-11-03T14:57:38Z

Lila at 14:55, 3 November 2021

2021-11-03T14:55:21Z

Lila at 14:53, 3 November 2021

2021-11-03T14:53:38Z

Lila at 14:52, 3 November 2021

2021-11-03T14:52:38Z

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 <p>Therefore, a monthly average of 600 talk-time minutes renders the plans equal.</li>
 </ol>
 ==Licensing==
 Content obtained and/or adapted from:
 * [https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-models-and-applications/ Models and Applications, Lumen Learning College Algebra] under a CC BY license

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 <li>Solve the equation. Be sure the solution can be explained in words including the units of measure.</li>
 </ol>
 ===Example: Setting Up a Linear Equation to Solve a Real-World Application===

@@ Line 62: / Line 62: @@
 <li>How many minutes of talk-time would yield equal monthly statements from both companies?</li>
 </ol>
-<div class="qa-wrapper" style="display: block"><span class="show-answer collapsed" style="cursor: pointer" data-target="q785384">Show Solution</span></p>
-<div id="q785384" class="hidden-answer" style="display: none">
 <ol>
 <li>The model for Company <em>A</em> can be written as <math>A=0.05x+34</math>. This includes the variable cost of <math>0.05x</math> plus the monthly service charge of $34. Company <em>B</em>’s package charges a higher monthly fee of $40, but a lower variable cost of <math>0.04x</math>. Company <em>B</em>’s model can be written as <math>B=0.04x+40</math>.</li>

@@ Line 54: / Line 54: @@
 </ol>
-<h3>Example: Modeling a Linear Equation to Solve an Unknown Number Problem</h3>
+===Example: Setting Up a Linear Equation to Solve a Real-World Application===
-<p>Find a linear equation to solve for the following unknown quantities: One number exceeds another number by <math>17</math> and their sum is <math>31</math>. Find the two numbers.</p>
+<p>There are two cell phone companies that offer different packages. Company A charges a monthly service fee of $34 plus $.05/min talk-time. Company B charges a monthly service fee of $40 plus $.04/min talk-time.</p>
+<ol>
-<p>Let <math>x</math> equal the first number. Then, as the second number exceeds the first by 17, we can write the second number as <math>x+17</math>. The sum of the two numbers is 31. We usually interpret the word <em>is</em> as an equal sign.</p>
+<li>Write a linear equation that models the packages offered by both companies.</li>
-<math>\begin{array}{ll}x+\left(x+17\right)=31 \\ 2x+17=31 & \text{Simplify and solve}. & \\ 2x=14 \\ x=7 & \\ & \\ x+17=7+17=24 \end{array}</math>
+<li>If the average number of minutes used each month is 1,160, which company offers the better plan?</li>
-<p>The two numbers are <math>7</math> and <math>24</math>.</p>
+<li>If the average number of minutes used each month is 420, which company offers the better plan?</li>
+<li>How many minutes of talk-time would yield equal monthly statements from both companies?</li>
 ==Licensing==
 Content obtained and/or adapted from:
 * [https://courses.lumenlearning.com/wmopen-collegealgebra/chapter/introduction-models-and-applications/ Models and Applications, Lumen Learning College Algebra] under a CC BY license

@@ Line 58: / Line 58: @@
 <p>Let <math>x</math> equal the first number. Then, as the second number exceeds the first by 17, we can write the second number as <math>x+17</math>. The sum of the two numbers is 31. We usually interpret the word <em>is</em> as an equal sign.</p>
-<math>\begin{array}{ll}x+\left(x+17\right)=31\hfill \\ 2x+17=31 & \text{Simplify and solve}. & \\ 2x=14 \\ x=7 & \\ & \\ x+17=7+17=24 \end{array}</math>
+<math>\begin{array}{ll}x+\left(x+17\right)=31 \\ 2x+17=31 & \text{Simplify and solve}. & \\ 2x=14 \\ x=7 & \\ & \\ x+17=7+17=24 \end{array}</math>
 <p>The two numbers are <math>7</math> and <math>24</math>.</p>

← Older revision		Revision as of 14:57, 3 November 2021
Line 53:		Line 53:
	<li>Solve the equation. Be sure the solution can be explained in words including the units of measure.</li>		<li>Solve the equation. Be sure the solution can be explained in words including the units of measure.</li>
	</ol>		</ol>
		+
		+	<h3>Example: Modeling a Linear Equation to Solve an Unknown Number Problem</h3>
		+	<p>Find a linear equation to solve for the following unknown quantities: One number exceeds another number by <math>17</math> and their sum is <math>31</math>. Find the two numbers.</p>
		+
		+	'''Solution''':
		+	<p>Let <math>x</math> equal the first number. Then, as the second number exceeds the first by 17, we can write the second number as <math>x+17</math>. The sum of the two numbers is 31. We usually interpret the word <em>is</em> as an equal sign.</p>
		+	<math>\begin{array}{ll}x+\left(x+17\right)=31\hfill \\ 2x+17=31\hfill&\text{Simplify and solve}.\hfill & \\ 2x=14\hfill \\ x=7\hfill & \\ \hfill & \\ x+17=7+17=24\hfill \end{array}</math>
		+	<p>The two numbers are <math>7</math> and <math>24</math>.</p>

@@ Line 1: / Line 1: @@
 <p>Many real-world applications can be modeled by linear equations. For example, a cell phone package may include a monthly service fee plus a charge per minute of talk-time; it costs a widget manufacturer a certain amount to produce <em>x </em>widgets per month plus monthly operating charges; a car rental company charges a daily fee plus an amount per mile driven. These are examples of applications we come across every day that are modeled by linear equations. In this section, we will set up and use linear equations to solve such problems.</p>
-<h2>Writing a Linear Equation to Solve an Application</h2>
 <p>To set up or model a linear equation to fit a real-world application, we must first determine the known quantities and define the unknown quantity as a variable. Then, we begin to interpret the words as mathematical expressions using mathematical symbols. Let us use the car rental example above. In this case, a known cost, such as $0.10/mi, is multiplied by an unknown quantity, the number of miles driven. Therefore, we can write <math>0.10x</math>. This expression represents a variable cost because it changes according to the number of miles driven.</p>
 <p>If a quantity is independent of a variable, we usually just add or subtract it according to the problem. As these amounts do not change, we call them fixed costs. Consider a car rental agency that charges $0.10/mi plus a daily fee of $50. We can use these quantities to model an equation that can be used to find the daily car rental cost <math>C</math>.</p>
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 </table>
 </blockquote>

@@ Line 10: / Line 10: @@
 <tr>
 <th>Verbal</th>
-<th><math>\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;</math> Translation to Math Operations</th>
+<th><math>\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;</math> Translation to Math Operations</th>
 </tr>
 <tr>
 <td>One number exceeds another by <em>a</em></td>
-<td><math>\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; x,\; x+a</math></td>
+<td><math>\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; x, \; x+a</math></td>
 </tr>
 <tr>
 <td>Twice a number</td>
-<td><math>\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;2x</math></td>
+<td><math>\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; 2x</math></td>
 </tr>
 <tr>
 <td>One number is <em>a </em>more than another number</td>
-<td><math>x,\text{ }x+a</math></td>
+<td><math>\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; x, \; x+a</math></td>
 </tr>
 <tr>
 <td>One number is <em>a </em>less than twice another number</td>
-<td><math>x,2x-a</math></td>
+<td><math>\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; x, \; 2x-a</math></td>
 </tr>
 <tr>
 <td>The product of a number and <em>a</em>, decreased by <em>b</em></td>
-<td><math>ax-b</math></td>
+<td><math>\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; ax-b</math></td>
 </tr>
 <tr>
 <td>The quotient of a number and the number plus <em>a </em>is three times the number</td>
-<td><math>\frac{x}{x+a}=3x</math></td>
+<td><math>\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \frac{x}{x+a}=3x</math></td>
 </tr>
 <tr>
 <td>The product of three times a number and the number decreased by <em>b </em>is <em>c</em></td>
-<td><math>3x\left(x-b\right)=c</math></td>
+<td><math>\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; 3x\left(x-b\right)=c</math></td>
 </tr>
 </table>