Question

1. circle all the points (x, y) that are solutions to the equation 4x − 6y = 12 show your work.

(3,4) (-1.5,1) (-2,0) (-4,3)
(handwritten work for (3,4): 4(3) -6=12; 12 +24=12; 12 -24=12; 12 ≠ 12)

Explanation:

Step1: Check (3,4)

Substitute \( x = 3 \), \( y = 4 \) into \( 4x - 6y \):
\( 4(3) - 6(4) = 12 - 24 = -12
eq 12 \), so not a solution.

Step2: Check (-1.5,1)

Substitute \( x = -1.5 \), \( y = 1 \) into \( 4x - 6y \):
\( 4(-1.5) - 6(1) = -6 - 6 = -12
eq 12 \), so not a solution.

Step3: Check (-2,0)

Substitute \( x = -2 \), \( y = 0 \) into \( 4x - 6y \):
\( 4(-2) - 6(0) = -8 - 0 = -8
eq 12 \), so not a solution.

Step4: Check (-4,3)

Substitute \( x = -4 \), \( y = 3 \) into \( 4x - 6y \):
\( 4(-4) - 6(3) = -16 - 18 = -34
eq 12 \), so not a solution. Wait, maybe there's a miscalculation? Wait, let's re - check. Wait, maybe the original work for (3,4) was wrong. Wait, \( 4x-6y = 12 \), let's solve for y: \( - 6y=12 - 4x\), \( y=\frac{4x - 12}{6}=\frac{2x - 6}{3}\).

Wait, for (3,4): \( y=\frac{2*3 - 6}{3}=\frac{6 - 6}{3}=0
eq4 \), so not a solution.

For (-1.5,1): \( x=-1.5=-\frac{3}{2}\), \( y=\frac{2*(-\frac{3}{2})-6}{3}=\frac{-3 - 6}{3}=\frac{-9}{3}=-3
eq1 \), not a solution.

For (-2,0): \( y=\frac{2*(-2)-6}{3}=\frac{-4 - 6}{3}=\frac{-10}{3}
eq0 \), not a solution.

For (-4,3): \( y=\frac{2*(-4)-6}{3}=\frac{-8 - 6}{3}=\frac{-14}{3}
eq3 \), not a solution. Wait, maybe the equation was written wrong? Or the points? Wait, maybe I made a mistake. Wait, the equation is \( 4x-6y = 12 \), let's rewrite it as \( 2x - 3y=6 \).

Wait, let's check (3,4) again: \( 23-34 = 6 - 12=-6
eq6 \). (-1.5,1): \( 2(-1.5)-31=-3 - 3=-6
eq6 \). (-2,0): \( 2(-2)-30=-4
eq6 \). (-4,3): \( 2(-4)-33=-8 - 9=-17
eq6 \). So none of the points are solutions? But that seems odd. Wait, maybe the original problem has a typo? Or my calculation is wrong. Wait, let's check the first point again. The original work in the image: \( 4(3)-6 = 12 \)? Wait, maybe the user made a mistake in writing the point? If the point was (3,0), then \( 4*3-6*0 = 12 \), which works. But as per the given points, none of them satisfy \( 4x - 6y=12 \).

Answer:

None of the given points \((3,4)\), \((-1.5,1)\), \((-2,0)\), \((-4,3)\) are solutions to the equation \(4x - 6y = 12\).