Question

1. solve the system of equations by graphing.

$y = 2x - 4$
$y = -x + 5$

Explanation:

Step1: Analyze \( y = 2x - 4 \)

For the line \( y = 2x - 4 \), the slope \( m = 2 \) and the y - intercept \( b=-4 \). To graph it, start at the point \( (0, - 4) \) (the y - intercept). Then, use the slope: from \( (0, - 4) \), move up 2 units and right 1 unit (since slope \(=\frac{\text{rise}}{\text{run}}=\frac{2}{1}\)) to get the next point \( (1, - 2) \), and so on.

Step2: Analyze \( y=-x + 5 \)

For the line \( y=-x + 5 \), the slope \( m=-1 \) and the y - intercept \( b = 5 \). Start at the point \( (0,5) \) (the y - intercept). Then, use the slope: from \( (0,5) \), move down 1 unit and right 1 unit (since slope \(=\frac{\text{rise}}{\text{run}}=\frac{- 1}{1}\)) to get the next point \( (1,4) \), and so on.

Step3: Find the intersection

When we graph both lines, we look for the point where they intersect. To find the intersection algebraically (to confirm), set the two equations equal to each other:
\(2x-4=-x + 5\)
Add \(x\) to both sides: \(2x+x-4=-x+x + 5\)
\(3x-4 = 5\)
Add 4 to both sides: \(3x-4 + 4=5 + 4\)
\(3x=9\)
Divide both sides by 3: \(x = 3\)
Substitute \(x = 3\) into \(y=-x + 5\): \(y=-3 + 5=2\)
So the intersection point (the solution to the system) is \( (3,2) \). When graphing, the two lines will cross at \( (3,2) \).

Answer:

The solution to the system of equations is \(x = 3\) and \(y = 2\), or the ordered pair \((3,2)\)