Question

1. solve the system by graphing:

$y = 2x + 1$
$y = -x + 4$

Explanation:

Step1: Analyze \( y = 2x + 1 \)

For \( y = 2x + 1 \), when \( x = 0 \), \( y = 1 \) (y - intercept). When \( y = 0 \), \( 0 = 2x + 1 \Rightarrow x = -\frac{1}{2} \). Plot points \((0, 1)\) and \((-\frac{1}{2}, 0)\), then draw the line.

Step2: Analyze \( y = -x + 4 \)

For \( y = -x + 4 \), when \( x = 0 \), \( y = 4 \) (y - intercept). When \( y = 0 \), \( 0 = -x + 4 \Rightarrow x = 4 \). Plot points \((0, 4)\) and \((4, 0)\), then draw the line.

Step3: Find Intersection

The two lines intersect where \( 2x + 1 = -x + 4 \). Solving: \( 2x + x = 4 - 1 \Rightarrow 3x = 3 \Rightarrow x = 1 \). Substitute \( x = 1 \) into \( y = 2x + 1 \): \( y = 2(1) + 1 = 3 \). So intersection is \((1, 3)\).

Answer:

The solution to the system is \( x = 1 \), \( y = 3 \) (or the ordered pair \((1, 3)\)).