To determine which graph matches the linear equation \(-2x + 3y = -6\), we can rewrite the equation in slope-intercept form \(y = mx + b\) or check if the given points satisfy the equation.
Let's express the equation in slope-intercept form:
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Start with the original equation: \[ -2x + 3y = -6 \]
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Solve for \(y\): \[ 3y = 2x - 6 \] \[ y = \frac{2}{3}x - 2 \]
Now we can check each of the answer options to see if the points satisfy the equation.
Checking the options:
A. (0, -4), (1, -10)
- For (0, -4): \[ y = \frac{2}{3}(0) - 2 = -2 \quad (\text{not } -4) \]
- For (1, -10): \[ y = \frac{2}{3}(1) - 2 = \frac{2}{3} - 2 = \frac{2}{3} - \frac{6}{3} = -\frac{4}{3} \quad (\text{not } -10) \]
B. (0, 2), (1, -1)
- For (0, 2): \[ y = \frac{2}{3}(0) - 2 = -2 \quad (\text{not } 2) \]
- For (1, -1): \[ y = \frac{2}{3}(1) - 2 = \frac{2}{3} - \frac{6}{3} = -\frac{4}{3} \quad (\text{not } -1) \]
C. (3, 0), (0, -2)
- For (3, 0): \[ 0 = \frac{2}{3}(3) - 2 = 2 - 2 = 0 \quad (\text{correct}) \]
- For (0, -2): \[ -2 = \frac{2}{3}(0) - 2 \quad (\text{correct}) \] Both points satisfy the equation.
D. (1, 2), (0, -4)
- For (1, 2): \[ 2 = \frac{2}{3}(1) - 2 = \frac{2}{3} - 2 \quad (\text{not } 2) \]
- For (0, -4): \[ -4 = \frac{2}{3}(0) - 2 \quad (\text{not } -4) \]
Conclusion
The correct answer is C. (3, 0), (0, -2) as both points satisfy the equation \(-2x + 3y = -6\).