Question

ch system by graphing.
-1 (10)
$y = 2x - 2$
$x + 3$
$x = 3$
$x - 4$ (12)
$y = x - 5$
$ - x + 4$
$y + 2x = 4$
$x + 4$ (14)
$y = 2x + 5$
$y = 8$
$y = \frac{1}{2}x - 1$
$ + 3$ (14)
$y = -2x + 2$
$4y = -8x + 16$

Explanation:

Problem 10: Solve \( y = 2x - 2 \) and \( x = 3 \) by Graphing

Step 1: Analyze \( x = 3 \)

The equation \( x = 3 \) is a vertical line passing through \( x = 3 \) on the x - axis.

Step 2: Analyze \( y = 2x - 2 \)

This is a linear equation in slope - intercept form \( y=mx + b \), where the slope \( m = 2 \) and the y - intercept \( b=-2 \). To find the point of intersection with \( x = 3 \), substitute \( x = 3 \) into \( y = 2x-2 \).
Substitute \( x = 3 \) into \( y = 2x - 2 \):
\( y=2\times3 - 2=6 - 2 = 4 \)
So the point of intersection is \( (3,4) \)

Step 1: Rewrite \( y + 2x=4 \)

Rewrite \( y + 2x = 4 \) in slope - intercept form \( y=mx + b \). We get \( y=-2x + 4 \), where the slope \( m=-2 \) and the y - intercept \( b = 4 \). The equation \( y=x - 5 \) has a slope \( m = 1 \) and a y - intercept \( b=-5 \)

Step 2: Find the intersection point

We can also find the intersection point algebraically (to verify the graph). Set the two equations equal to each other:
\( x-5=-2x + 4 \)
Add \( 2x \) to both sides: \( x+2x-5=-2x + 2x+4\Rightarrow3x-5 = 4 \)
Add 5 to both sides: \( 3x-5 + 5=4 + 5\Rightarrow3x=9 \)
Divide both sides by 3: \( x = 3 \)
Substitute \( x = 3 \) into \( y=x - 5 \): \( y=3-5=-2 \)
So the point of intersection is \( (3,-2) \)

Step 1: Analyze the two lines

The line \( y = 2x+5 \) has a slope \( m = 2 \) and a y - intercept \( b = 5 \). The line \( y=\frac{1}{2}x-1 \) has a slope \( m=\frac{1}{2} \) and a y - intercept \( b=-1 \)

Step 2: Find the intersection point

Set \( 2x + 5=\frac{1}{2}x-1 \)
Subtract \( \frac{1}{2}x \) from both sides: \( 2x-\frac{1}{2}x+5=\frac{1}{2}x-\frac{1}{2}x-1\Rightarrow\frac{4x - x}{2}+5=-1\Rightarrow\frac{3x}{2}+5=-1 \)
Subtract 5 from both sides: \( \frac{3x}{2}+5 - 5=-1 - 5\Rightarrow\frac{3x}{2}=-6 \)
Multiply both sides by \( \frac{2}{3} \): \( x=-6\times\frac{2}{3}=-4 \)
Substitute \( x = - 4 \) into \( y = 2x+5 \): \( y=2\times(-4)+5=-8 + 5=-3 \)
So the point of intersection is \( (-4,-3) \)

Answer:

The solution of the system \(

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

\) is \( (3,4) \)

Problem 12: Solve \( y=x - 5 \) and \( y + 2x=4 \) by Graphing