Question

solve each system using a graphing calculator.

1. $y = \frac{3}{4}x + 2$

$y = x + 1$

1. $2x + y = 4$

$x - y = 2$

1. $2x - 5y = 10$

$6x - 15y = 75$

1. $3x + 2y = 4$

$-x + 3y = -5$

Explanation:

Problem 26:

Step1: Analyze the equations

We have two linear equations: \( y = \frac{3}{4}x + 2 \) and \( y = x + 1 \). To solve the system, we can set them equal to each other since both equal \( y \).

Step2: Solve for \( x \)

\( \frac{3}{4}x + 2 = x + 1 \)
Subtract \( \frac{3}{4}x \) from both sides: \( 2 = \frac{1}{4}x + 1 \)
Subtract 1 from both sides: \( 1 = \frac{1}{4}x \)
Multiply both sides by 4: \( x = 4 \)

Step3: Solve for \( y \)

Substitute \( x = 4 \) into \( y = x + 1 \): \( y = 4 + 1 = 5 \)

Step1: Rewrite the equations in slope - intercept form

For \( 2x + y = 4 \), we get \( y=-2x + 4 \). For \( x - y = 2 \), we can rewrite it as \( y=x - 2 \)

Step2: Find the intersection

To find the intersection, we can set \( -2x + 4=x - 2 \)
Add \( 2x \) to both sides: \( 4 = 3x-2 \)
Add 2 to both sides: \( 6 = 3x \)
Divide by 3: \( x = 2 \)

Step3: Find \( y \)

Substitute \( x = 2 \) into \( y=x - 2 \), we get \( y=2 - 2=0 \)

Step1: Analyze the equations

First, simplify the second equation. Divide \( 6x-15y = 75 \) by 3, we get \( 2x-5y = 25 \)
The first equation is \( 2x-5y = 10 \)

Step2: Determine the nature of the system

The two equations \( 2x - 5y=10 \) and \( 2x - 5y = 25 \) are parallel (same slope, different y - intercepts) because they are of the form \( Ax+By = C \) with \( A = 2 \), \( B=-5 \) and different \( C \) values (\( C_1 = 10 \), \( C_2 = 25 \))

Answer:

The solution is \( (4, 5) \)

Problem 27: