Question

skill #5: i can solve systems of linear equations by graphing. solve each system by graphing. a) $y = -\frac{1}{3}x + 2$ $y = -2x - 3$ c) $2x + y = -4$ $x + 4y = 12$

Explanation:

Part (a)

Step 1: Analyze the first equation \( y = -\frac{1}{3}x + 2 \)

This is in slope - intercept form \( y=mx + b \), where the slope \( m=-\frac{1}{3} \) and the y - intercept \( b = 2 \). To graph this line, we start by plotting the y - intercept at \( (0,2) \). Then, using the slope (rise over run), from the point \( (0,2) \), we move down 1 unit and right 3 units (or up 1 unit and left 3 units) to get other points on the line.

Step 2: Analyze the second equation \( y=-2x - 3 \)

This is also in slope - intercept form with slope \( m=-2 \) and y - intercept \( b=-3 \). We plot the y - intercept at \( (0, - 3) \). Then, using the slope, from the point \( (0,-3) \), we move down 2 units and right 1 unit (or up 2 units and left 1 unit) to get other points on the line.

Step 3: Find the intersection point

When we graph both lines, we look for the point where they cross each other. By solving the system algebraically (to confirm the intersection point: set \( -\frac{1}{3}x + 2=-2x - 3 \)
\( -\frac{1}{3}x+2x=-3 - 2 \)
\( \frac{-1 + 6}{3}x=-5 \)
\( \frac{5}{3}x=-5 \)
\( x=-3 \)
Substitute \( x = - 3 \) into \( y=-2x - 3 \), we get \( y=-2\times(-3)-3=6 - 3 = 3 \))
The two lines intersect at \( (-3,3) \)

Part (c)

Step 1: Rewrite the first equation in slope - intercept form

Given \( 2x + y=-4 \), we can rewrite it as \( y=-2x - 4 \). Here, the slope \( m=-2 \) and the y - intercept \( b=-4 \). We plot the y - intercept at \( (0,-4) \) and use the slope to find other points (move down 2 units and right 1 unit or up 2 units and left 1 unit).

Step 2: Rewrite the second equation in slope - intercept form

Given \( x + 4y=12 \), we solve for \( y \):
\( 4y=-x + 12 \)
\( y=-\frac{1}{4}x+3 \)
The slope \( m = -\frac{1}{4} \) and the y - intercept \( b = 3 \). We plot the y - intercept at \( (0,3) \) and use the slope (move down 1 unit and right 4 units or up 1 unit and left 4 units) to find other points.

Step 3: Find the intersection point

To find the intersection point algebraically (to confirm), we solve the system:
From \( y=-2x - 4 \) and \( y=-\frac{1}{4}x + 3 \)
Set \( -2x-4=-\frac{1}{4}x + 3 \)
\( -2x+\frac{1}{4}x=3 + 4 \)
\( \frac{-8x+x}{4}=7 \)
\( \frac{-7x}{4}=7 \)
\( -7x = 28 \)
\( x=-4 \)
Substitute \( x=-4 \) into \( y=-2x - 4 \), we get \( y=-2\times(-4)-4=8 - 4=4 \)
The two lines intersect at \( (-4,4) \)

Answer:

s:
a) The solution of the system \(

$$\begin{cases}y = -\frac{1}{3}x + 2\\y=-2x - 3\end{cases}$$

\) is \( x=-3,y = 3 \) or the ordered pair \( (-3,3) \)

c) The solution of the system \(

$$\begin{cases}2x + y=-4\\x + 4y=12\end{cases}$$

\) is \( x=-4,y = 4 \) or the ordered pair \( (-4,4) \)