Problem 27 Solve a System of Linear Equatio... [FREE SOLUTION] (2024)

Table of Contents

Short Answer Step by step solution Write the equations Convert the equations to slope-intercept form Convert the second equation to slope-intercept form Graph the first equation Graph the second equation Find the intersection of the lines Verify the solution Key Concepts One App. One Place for Learning. Most popular questions from this chapter Recommended explanations on Math Textbooks Discrete Mathematics Applied Mathematics Mechanics Maths Geometry Decision Maths Logic and Functions Study anywhere. Anytime. Across all devices. FAQs References

Get started for free

Get started for free Log out

Chapter 5: Problem 27

Solve a System of Linear Equations by Graphing In the following exercises,solve the following systems of equations by graphing. $$ \left\\{\begin{array}{l} x+y=-4 \\ -x+2 y=-2 \end{array}\right. $$

Short Answer

Expert verified

The solution is $ (-2, -2) $.

Step by step solution

Write the equations

Given the system of equations: $ x + y = -4 $ $ -x + 2y = -2 $

Convert the equations to slope-intercept form

$ x + y = -4 $ Subtract $ x $ from both sides: $ y = -x - 4 $

Convert the second equation to slope-intercept form

$ -x + 2y = -2 $ Add $ x $ to both sides: $ 2y = x - 2 $ Divide both sides by 2: $ y = \frac{1}{2}x - 1 $

Graph the first equation

Graph the equation $ y = -x - 4 $. Start at the y-intercept $ (0, -4) $. Use the slope to plot another point: From $ (0, -4) $, go down 1 unit and right 1 unit to point $ (1, -5) $.

Graph the second equation

Graph the equation $ y = \frac{1}{2}x - 1 $. Start at the y-intercept $ (0, -1) $. Use the slope to plot another point: From $ (0, -1) $, go up 1 unit and right 2 units to point $ (2, 0) $.

Find the intersection of the lines

The intersection of the two graphs represents the solution to the system of equations. The lines intersect at the point $ (-2, -2) $.

Verify the solution

Substitute $ x = -2 $ and $ y = -2 $ into both original equations to verify: For $ x + y = -4 $: $ -2 + (-2) = -4 $ which is true. For $ -x + 2y = -2 $: $ -(-2) + 2(-2) = -2 $ which is true.

Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations

A linear equation is a type of equation that creates a straight line when graphed on a coordinate plane. It has the general form of $Ax + By = C$, where $A$, $B$, and $C$ are constants. In a system of linear equations, you have two or more linear equations that you deal with at the same time. For example, in the given exercise, the system consists of two linear equations:

$x + y = -4$
$-x + 2y = -2$

To solve a system of linear equations by graphing, you graph each equation on the same set of axes and look for the point where the lines intersect.

Graphing Lines

When you graph a linear equation, you convert it into a format that easily shows its characteristics. The most common form for this is the slope-intercept form, $y = mx + b$, where $m$ represents the slope and $b$ represents the y-intercept. For the given exercise, we converted both equations:

$x + y = -4$ becomes $y = -x - 4$
$-x + 2y = -2$ becomes $y = \frac{1}{2}x - 1$

Once they are in slope-intercept form, you can easily plot the y-intercepts and use the slopes to find other points on each line.

Intersection of Lines

The intersection of two lines is the point where they cross each other. This point represents the solution to the system of equations because it is the only point that satisfies both equations simultaneously. In the exercise, after graphing the equations $y = -x - 4$ and $y = \frac{1}{2}x - 1$, we found they intersect at $(-2, -2)$. This implies that $x = -2$ and $y = -2$ is the solution to the system.

Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis, which means the value of $x$ at this point is zero. For example, in the equation $y = -x - 4$, the y-intercept is $ -4 $, so the line crosses the y-axis at (0, -4). In the equation $y = \frac{1}{2}x - 1$, the y-intercept is $ -1 $, so the line crosses the y-axis at (0, -1). Starting with the y-intercept makes it easier to draw the line on the coordinate plane.

Slope

The slope of a line describes how steep the line is and the direction it goes (up or down). It is calculated as the change in $y$ over the change in $x$ ($\frac{ \text{rise} }{ \text{run} }$). In the slope-intercept form $y = mx + b$, $m$ represents the slope. A positive slope means the line inclines upwards, while a negative slope means it declines downward. In the exercise:

For $y = -x - 4$, the slope is $ -1 $.
For $y = \frac{1}{2}x - 1$, the slope is $ \frac{1}{2} $.

Using the slope, you can plot additional points starting from the y-intercept to ensure accuracy when graphing the line.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

In the following exercises, determine whether each ordered pair is a solutionto the system. $\left\\{\begin{array}{l}6 x-5 y<20 \\ -2 x+7 y>-8\end{array}\right.$ (a) (1,-3) (b) (-4,4)In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} y \geq-\frac{2}{3} x+2 \\ y>2 x-3 \end{array}\right. $$In the following exercises, determine whether each ordered pair is a solutionto the system. $\left\\{\begin{array}{l}4 x-y<10 \\ -2 x+2 y>-8\end{array}\right.$ (a) (5,-2) (b) (-1,3)In the following exercises, solve each system by graphing. $$ \left\\{\begin{array}{l} x-3 y>4 \\ y \leq-1 \end{array}\right. $$At a school concert, 425 tickets were sold. Student tickets cost $\$ 5$ eachand adult tickets cost $\$ 8$ each. The total receipts for the concert were$\$ 2,851$. Solve the system $\left\\{\begin{array}{l}s+a=425 \\ 5 s+8 a=2,851\end{array}\right.$ to find $s,$ the number of student tickets and $a,$ the number of adult tickets.

See all solutions

Recommended explanations on Math Textbooks

Discrete Mathematics

Read Explanation

Applied Mathematics

Read Explanation

Mechanics Maths

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation

Logic and Functions

Read Explanation

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.

Sign-up for free

This website uses cookies to improve your experience. We'll assume you're ok with this, but you can opt-out if you wish. Accept

Privacy & Cookies Policy

Privacy Overview

This website uses cookies to improve your experience while you navigate through the website. Out of these, the cookies that are categorized as necessary are stored on your browser as they are essential for the working of basic functionalities of the website. We also use third-party cookies that help us analyze and understand how you use this website. These cookies will be stored in your browser only with your consent. You also have the option to opt-out of these cookies. But opting out of some of these cookies may affect your browsing experience.

Necessary

Always Enabled

Necessary cookies are absolutely essential for the website to function properly. This category only includes cookies that ensures basic functionalities and security features of the website. These cookies do not store any personal information.

Non-necessary

Any cookies that may not be particularly necessary for the website to function and is used specifically to collect user personal data via analytics, ads, other embedded contents are termed as non-necessary cookies. It is mandatory to procure user consent prior to running these cookies on your website.

SAVE & ACCEPT

Problem 27 Solve a System of Linear Equatio... [FREE SOLUTION] (2024)

FAQs

How do you solve a problem solving linear equation? ›

The steps for solving linear equations are:

Simplify both sides of the equation and combine all same-side like terms.
Combine opposite-side like terms to obtain the variable term on one side of the equal sign and the constant term on the other.
Divide or multiply as needed to isolate the variable.
Check the answer.

Oct 6, 2021

Read On ›

What is a real life example of a linear equation in two variables? ›

Suppose we rent a car with a charge of $200 plus $25 for every hour. Here you don't know how many hours you will travel so by using "t" to represent the number of hours to your destination and "x" to represent the cost of that taxi ride, this can be framed in an equation as x = 25 × t + 200.

Read The Full Story ›

What is an example of a system of linear equations? ›

The system of linear equations in two variables is the set of equations that contain only two variables. For example, 2x + 3y = 4; 3x + 5y = 12 are the system of equations in two variables. There are several methods of solving linear equations in two variables, such as: Graphical method.

Get More Info ›

What is an example of a linear equation and a solution? ›

The linear equations in one variable is an equation which is expressed in the form of ax+b = 0, where a and b are two integers, and x is a variable and has only one solution. For example, 2x+3=8 is a linear equation having a single variable in it. Therefore, this equation has only one solution, which is x = 5/2.

Get More Info Here ›

What is the easiest way to solve a linear equation? ›

To solve linear equations, find the value of the variable that makes the equation true. Use the inverse of the number that multiplies the variable, and multiply or divide both sides by it. Simplify the result to get the variable value. Check your answer by plugging it back into the equation.

Discover More ›

What are 4 examples of linear equations? ›

Some of the examples of linear equations are 2x – 3 = 0, 2y = 8, m + 1 = 0, x/2 = 3, x + y = 2, 3x – y + z = 3.

Discover More ›

What are 5 examples of linear equation in two variables? ›

Examples

2 x + y = 7 \displaystyle 2x+y=7 2x+y=7.
x − 3 y = 6 \displaystyle x-3y=6 x−3y=6.
2 x − 5 y = 4 \displaystyle 2x-5y=4 2x−5y=4.
2 ( x − 3 ) − ( 2 y − 1 ) = 1 \displaystyle 2(x-3)-(2y-1)=1 2(x−3)−(2y−1)=1.

Keep Reading ›

What is the formula for a linear equation? ›

The slope-intercept form of a linear equation is y = mx + b. In the equation, x and y are the variables. The numbers m and b give the slope of the line (m) and the value of y when x is 0 (b).

Keep Reading ›

What is a linear system of equations in real life? ›

You can use a linear equation to depict almost any circ*mstance involving an unknown number, such as estimating income over time, computing mileage rates, or predicting profit. Many people use linear equations on a daily basis, even if they don't visualize a line graph in their heads.

Tell Me More ›

What are the three ways to solve a system of equations? ›

There are three ways to solve a system of linear equations: graphing, substitution, and elimination. The solution to a system of linear equations is the ordered pair (or pairs) that satisfies all equations in the system. The solution is the ordered pair(s) common to all lines in the system when the lines are graphed.

Get More Info Here ›

What are the methods for solving linear equations? ›

To solve a linear equation in two variables, any of the above-mentioned methods can be used i.e. graphical method, elimination method, substitution method, cross multiplication method, matrix method, determinants method.

How do you solve a linear function equation? ›

Solving Linear Functions. A linear function is a function with the form f(x) = ax' + b. It looks like a regular linear equation, but instead of using y, the linear function notation is f(x). To solve a linear function, you would be given the value of f(x) and be asked to find x.

View Details ›

What are the five steps to solving a linear equation? ›

Solve linear equations using a general strategy.

Simplify each side of the equation as much as possible. ...
Collect all the variable terms on one side of the equation. ...
Collect all the constant terms on the other side of the equation. ...
Make the coefficient of the variable term equal to 1. ...
Check the solution.

May 6, 2020

Explore More ›

What is the rule to solve linear equations? ›

If given a linear equation of the form ax+b=c, then we can solve it in two steps. First, use the appropriate equality property of addition or subtraction to isolate the variable term. Next, isolate the variable using the equality property of multiplication or division.

Show Me More ›

What is the formula for a linear problem? ›

The standard form of a linear equation in one variable is of the form Ax + B = 0. Here, x is a variable, A is a coefficient and B is constant. The standard form of a linear equation in two variables is of the form Ax + By = C. Here, x and y are variables, A and B are coefficients and C is a constant.