Question

Solve the system of equations graphed on the coordinate axes below.
y, equals, minus, start fraction, 2, divided by, 3, end fraction, x, plus, 2
y=
−
3
2
​
x+2
y, equals, one third, x, minus, 4
y=
3
1
​
x−4

Answer 1

To solve the system of equations given by:

1. \( y = -\frac{2}{3}x + 2 \)
2. \( y = \frac{1}{3}x - 4 \)

we can set them equal to each other since both represent \( y \):

\[
-\frac{2}{3}x + 2 = \frac{1}{3}x - 4
\]

Next, we can eliminate the fractions by multiplying every term by 3:

\[
3\left(-\frac{2}{3}x\right) + 3(2) = 3\left(\frac{1}{3}x\right) - 3(4)
\]
\[
-2x + 6 = x - 12
\]

Now, we can rearrange the equation to isolate \( x \):

Add \( 2x \) to both sides:

\[
6 = 3x - 12
\]

Next, add 12 to both sides:

\[
6 + 12 = 3x
\]
\[
18 = 3x
\]

Now, divide by 3:

\[
x = 6
\]

Now that we have the \( x \) value, we can substitute it back into either equation to find the \( y \) value. We'll use the first equation:

\[
y = -\frac{2}{3}(6) + 2
\]
\[
y = -4 + 2
\]
\[
y = -2
\]

Thus, the solution to the system of equations is:

\[
(x, y) = (6, -2)
\]

So, the coordinates of the intersection point are \((6, -2)\).