Question

Type an equation that has the same slope as 2x + 3y = 8 and the same y-intercept as -3 - 4y + 2 = 0. Solve for this equation and determine the slope and y-intercept.

a) 2x - 3y = 8
b) 4x + 2y = -3
c) 2x + 4y = 8
d) -2x + 3y = -8

Answer 1

The equation with the same slope and y-intercept as the given equations is y = (-2/3)x + 8/3, with a slope of -2/3 and a y-intercept of 8/3.

Step-by-step explanation:

Slope and y-intercept of a Linear Equation

To find an equation with the same slope as 2x + 3y = 8, we can rearrange the equation to solve for y: 3y = -2x + 8. Dividing both sides by 3 gives y = (-2/3)x + 8/3. Therefore, the equation with the same slope is y = (-2/3)x + 8/3.

To find an equation with the same y-intercept as -3 - 4y + 2 = 0, we need to isolate the y-term: -4y = -3 - 2. Simplifying gives -4y = -5, and dividing both sides by -4 gives y = 5/4. Therefore, the equation with the same y-intercept is y = 5/4.

Putting it all together, the equation with the same slope and y-intercept is y = (-2/3)x + 8/3, which has a slope of -2/3 and a y-intercept of 8/3.

Answer 2

d) -2x + 3y = -8.he y-intercept of -2x + 3y = -8, derived by substituting the y-intercept of -1.25 into the equation, aligns with the y-intercept (-1.25) calculated from -3 - 4y + 2 = 0. Consequently, the equation -2x + 3y = -8 satisfies both conditions, ensuring an equivalent slope and y-intercept as required.

Explanation

The equation with the same slope as 2x + 3y = 8 is -2x + 3y = -8. To find the equation with the same y-intercept as -3 - 4y + 2 = 0, rearrange it to solve for y, resulting in y = -1.25.

Then, substitute the y-intercept (-1.25) into the equation with the same slope to solve for the y-intercept, which gives -2x + 3(-1.25) = -8, simplifying to -2x - 3.75 = -8, and further simplification yields -2x = -4.25, giving the x-intercept as 2.125. Hence, the equation -2x + 3y = -8 has the same slope as 2x + 3y = 8 and the same y-intercept as -3 - 4y + 2 = 0.

The equation -2x + 3y = -8 shares the slope of 2x + 3y = 8 because they have the same coefficient for x and y terms, and they differ only in the constant term. Additionally, the y-intercept of -2x + 3y = -8, derived by substituting the y-intercept of -1.25 into the equation, aligns with the y-intercept (-1.25) calculated from -3 - 4y + 2 = 0. Consequently, the equation -2x + 3y = -8 satisfies both conditions, ensuring an equivalent slope and y-intercept as required.