Question

directions: write the system of equations shown on the graph and identify its solution.
1.

solution:
2.

solution:
directions: solve each system by graphing. be sure to clearly give the solution.

1. \\(\

$$\begin{cases} y = -x - 4 \\\\ y = 5x + 2 \\end{cases}$$

\\)

1. \\(\

$$\begin{cases} y = \\frac{2}{3}x + 3 \\\\ y = -x - 7 \\end{cases}$$

\\)

1. \\(\

$$\begin{cases} y = \\frac{1}{2}x + 4 \\\\ y = -\\frac{5}{2}x - 2 \\end{cases}$$

\\)

1. \\(\

$$\begin{cases} y = -2x + 7 \\\\ y = 5 \\end{cases}$$

\\)

Explanation:

Problem 3

Step1: Analyze the first equation \( y = -x - 4 \)

This is a linear equation in slope - intercept form \( y=mx + b \), where the slope \( m=- 1 \) and the y - intercept \( b=-4 \). To graph it, we can find two points. When \( x = 0 \), \( y=-4 \). When \( y = 0 \), \( 0=-x - 4\), so \( x=-4 \). So two points on this line are \( (0,-4) \) and \( (-4,0) \).

Step2: Analyze the second equation \( y = 5x+2 \)

This is also in slope - intercept form with slope \( m = 5 \) and y - intercept \( b = 2 \). When \( x = 0 \), \( y = 2 \). When \( x=-1 \), \( y=5\times(-1)+2=-3 \). So two points on this line are \( (0,2) \) and \( (-1,-3) \).

Step3: Find the intersection point

To find the solution of the system, we can set the two equations equal to each other:
\( -x - 4=5x + 2 \)
Add \( x \) to both sides: \( - 4=6x + 2 \)
Subtract 2 from both sides: \( -6 = 6x \)
Divide both sides by 6: \( x=-1 \)
Substitute \( x = - 1 \) into \( y=-x - 4 \), we get \( y=-(-1)-4=1 - 4=-3 \)
We can also find the intersection by graphing the two lines. The two lines intersect at the point \( (-1,-3) \)

Step1: Analyze the first equation \( y=\frac{2}{3}x + 3 \)

It is in slope - intercept form \( y = mx + b \), where \( m=\frac{2}{3} \) and \( b = 3 \). When \( x = 0 \), \( y = 3 \). When \( x = 3 \), \( y=\frac{2}{3}\times3+3=2 + 3=5 \). So two points are \( (0,3) \) and \( (3,5) \)

Step2: Analyze the second equation \( y=-x - 7 \)

In slope - intercept form, \( m=-1 \), \( b=-7 \). When \( x = 0 \), \( y=-7 \). When \( x=-7 \), \( y = 0 \). So two points are \( (0,-7) \) and \( (-7,0) \)

Step3: Find the intersection point

Set \( \frac{2}{3}x+3=-x - 7 \)
Multiply through by 3 to clear the fraction: \( 2x + 9=-3x-21 \)
Add \( 3x \) to both sides: \( 5x+9=-21 \)
Subtract 9 from both sides: \( 5x=-30 \)
Divide by 5: \( x=-6 \)
Substitute \( x = - 6 \) into \( y=-x - 7 \), \( y=-(-6)-7=6 - 7=-1 \)
By graphing, the two lines intersect at \( (-6,-1) \)

Step1: Analyze the first equation \( y=\frac{1}{2}x + 4 \)

Slope - intercept form with \( m=\frac{1}{2} \), \( b = 4 \). When \( x = 0 \), \( y = 4 \). When \( x = 2 \), \( y=\frac{1}{2}\times2+4=1 + 4=5 \). Points: \( (0,4) \), \( (2,5) \)

Step2: Analyze the second equation \( y=-\frac{5}{2}x-2 \)

Slope - intercept form with \( m=-\frac{5}{2} \), \( b=-2 \). When \( x = 0 \), \( y=-2 \). When \( x=-2 \), \( y=-\frac{5}{2}\times(-2)-2=5 - 2=3 \). Points: \( (0,-2) \), \( (-2,3) \)

Step3: Find the intersection point

Set \( \frac{1}{2}x + 4=-\frac{5}{2}x-2 \)
Multiply through by 2: \( x + 8=-5x-4 \)
Add \( 5x \) to both sides: \( 6x+8=-4 \)
Subtract 8 from both sides: \( 6x=-12 \)
Divide by 6: \( x=-2 \)
Substitute \( x=-2 \) into \( y=\frac{1}{2}x + 4 \), \( y=\frac{1}{2}\times(-2)+4=-1 + 4=3 \)
The lines intersect at \( (-2,3) \)

Answer:

The solution of the system \(

$$\begin{cases}y=-x - 4\\y = 5x+2\end{cases}$$

\) is \( x=-1,y=-3 \) or the ordered pair \( (-1,-3) \)

Problem 4