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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

01

## Write the equations

Given the system of equations: \( x + y = -4 \) \( -x + 2y = -2 \)

02

## Convert the equations to slope-intercept form

\( x + y = -4 \) Subtract \( x \) from both sides: \( y = -x - 4 \)

03

## Convert the second equation to slope-intercept form

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

04

## 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 \( (1, -5) \).

05

## 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 \( (2, 0) \).

06

## 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 \( (-2, -2) \).

07

## 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.

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