Let's analyze the equation \(3x - 2y = 18\) step by step.
-
Convert to slope-intercept form (y = mx + b): The goal is to solve for \(y\):
\[ 3x - 2y = 18 \]
Subtract \(3x\) from both sides:
\[ -2y = -3x + 18 \]
Divide by \(-2\):
\[ y = \frac{3}{2}x - 9 \]
Here we see that the slope \(m\) is \(\frac{3}{2}\) and the y-intercept \(b\) is \(-9\).
Conclusion: The equation is indeed a linear function in slope-intercept form, so this statement is correct.
-
Check for y-intercept and x-intercept:
-
The y-intercept (when \(x = 0\)) is: \[ y = \frac{3}{2}(0) - 9 = -9 \] Hence, the y-intercept is \((0, -9)\).
-
The x-intercept (when \(y = 0\)): \[ 0 = \frac{3}{2}x - 9 \] Transport the \(-9\) to the other side: \[ \frac{3}{2}x = 9 \] Now multiply by \(\frac{2}{3}\): \[ x = 6 \] Thus, the x-intercept is \((6, 0)\).
Conclusion: This statement is also correct.
-
-
Check the point-slope form: The point-slope form generally looks like \(y - y_1 = m(x - x_1)\). The conversion to any specific form wasn't requested; however, since authentic point-slope form involves the slope and either an intercept (the y-intercept would be relevant here), it appears they provided the wrong scenario for a conversion claim here.
From our evaluation:
- The equation is a linear function in slope-intercept form.
- The y-intercept is \((0, -9)\) and the x-intercept is \((6, 0)\).
Final Verification: The two correct statements are:
- "This is a linear function in slope-intercept form."
- "The y-intercept is \((0, -9)\), and the x-intercept is \((6, 0)\)."
Overall, based on your initial statements, only the point-slope form phrase seems misdirected and can be ruled out as incorrect. Hence the correct answers would indeed be the first and third statements.