To determine which graph represents the equation \( 4x - 3y = 12 \), we can rearrange this equation into slope-intercept form (i.e., \( y = mx + b \)).
Starting with: \[ 4x - 3y = 12 \] we can isolate \( y \): \[ -3y = -4x + 12 \] \[ y = \frac{4}{3}x - 4 \]
From this, we can see that the y-intercept (\(b\)) is \(-4\), meaning the line crosses the y-axis at \((0, -4)\).
Next, we can find another point on the line by choosing an \(x\) value. Let's use \(x = 3\): \[ y = \frac{4}{3}(3) - 4 = 4 - 4 = 0 \] So, the point \((3, 0)\) is also on the line.
Now, we can check which of the provided graph options contains the points \((0, -4)\) and \((3, 0)\):
- A coordinate plane with a line passing through (0, negative 3) and (1, 1): This option does not include \((0, -4)\).
- A coordinate plane with a line passing through (negative 2, negative 1) and (4, 1): This option does not include \((0, -4)\).
- A coordinate plane with a line passing through (0, negative 12) and (3, 0): This option does not include \((0, -4)\).
- A coordinate plane with a line passing through (0, negative 4) and (3, 0): This option correctly includes both \((0, -4)\) and \((3, 0)\).
Thus, the correct graph is the fourth option, which passes through \((0, -4)\) and \((3, 0)\).