Compare the slopes and y- intercepts of the graphs of the equations in the linear system to determine whether the system has one solution, no solution, or infinitely many solutions.

Question

1. x = -3y + 28

x + 4y =36

1. 2x + 3y = 11

-4x - 6y = -22

Answer 1

Let's call the 4 equations A, B, C, and D just so we can be clear.

Equation A:

x = -3y + 28 [add 3y to both sides to get:

3y + x = 28 [subtract x from both sides to get:

3y = -x + 28 [divide both sides by 3 to get:

y = -1/3x + 28/3 so the slope is -1/3 and the y-intercept is 28/3

Equation B:

x + 4y = 36 [subtract x from both sides to get:

4y = -x + 36 [divide both sides by 4 to get:

y = -1/4x + 9 so the slope is -1/4 and the y-intercept is 9

Comparing Equations A and B, we have different slopes which means there will be one solution or intersection point.

Equation C:

2x + 3y = 11 [subtract 2x from both sides to get:

3y = -2x + 11 [divide both sides by 3 to get:

y = -2/3x + 11/3 so the slope is -2/3 and the y-intercept is 11/3

Equation D:

-4x - 6y = -22 [add 4x to both sides to get:

-6y = 4x - 22 [divide both sides by -6 to get:

y = -4/6x + 22/6 [reduce the fractions to get:

y = -2/3x + 11/3 so the slope is -2/3 and the y-intercept is 11/3

Comparing Equations C and D, we have the same slope (-2/3) and the same y-intercept (11/3) so these two lines lie right on top of each other meaning there are infinitely many solutions.