To solve the system of equations given by:
- \( y = -\frac{2}{3}x + 2 \)
- \( y = \frac{1}{3}x - 4 \)
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 \((6, -2)\).