Question

state how many solutions there will be for each system.

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

$y = \frac{2}{3}x - 2$

1. $y = 7 - 3x$

$y = 7 + 3x$

1. $2y = 6x + 8$

$y = 3x + 6$

1. $y = \frac{1}{2}(x - 6)$

$y = \frac{1}{2}(x + 2) - 4$

1. $9x = 18$

$3x = 12$

1. $y = 4x - 5$

$8x + y = 3$

1. two airplanes mapped their courses on a coordinate grid. the first plane followed the path of $y = 3x - 8$. the second airplane flew its daily flight on the path $y = \frac{1}{2}(6x - 16)$. describe the similarities or differences in the planes’ paths.

Explanation:

Problem 4

Step1: Analyze slopes and y-intercepts

The two equations are \( y = \frac{2}{3}x + 2 \) and \( y = \frac{2}{3}x - 2 \). Both have the same slope (\( m = \frac{2}{3} \)) but different y-intercepts (2 and -2).

Step2: Determine number of solutions

Parallel lines (same slope, different y-intercepts) never intersect, so there are no solutions.

Step1: Rewrite equations in slope-intercept form

First equation: \( y = -3x + 7 \) (slope \( m = -3 \), y-intercept 7). Second equation: \( y = 3x + 7 \) (slope \( m = 3 \), y-intercept 7).

Step2: Determine number of solutions

Different slopes mean the lines intersect at one point, so there is 1 solution.

Step1: Simplify the first equation

Divide \( 2y = 6x + 8 \) by 2: \( y = 3x + 4 \). Second equation: \( y = 3x + 6 \).

Step2: Analyze slopes and y-intercepts

Same slope (\( m = 3 \)) but different y-intercepts (4 and 6). Parallel lines, so no solutions.

Answer:

0 solutions

Problem 5