To analyze the function given by the equation \(6x - 5y = -10\), let's convert it into slope-intercept form and find the intercepts.
Step 1: Convert to Slope-Intercept Form
The slope-intercept form of a line is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.
Starting with: \[ 6x - 5y = -10 \]
Rearranging to solve for \(y\): \[ -5y = -6x - 10 \]
Dividing everything by -5: \[ y = \frac{6}{5}x + 2 \]
Now we can see that:
- The slope \(m\) is \(\frac{6}{5}\).
- The y-intercept \(b\) is \(2\), so the y-intercept is \((0, 2)\).
Step 2: Find the x-intercept
To find the x-intercept, set \(y = 0\) in the original equation: \[ 6x - 5(0) = -10 \implies 6x = -10 \implies x = -\frac{10}{6} = -\frac{5}{3} \]
So the x-intercept is \((- \frac{5}{3}, 0)\).
Conclusion
Now we can analyze the statements:
-
The y-intercept is (0, 2), and the x-intercept is (-3/5, 0).
- The y-intercept is correct, but the x-intercept calculated is \((- \frac{5}{3}, 0)\), not \((- \frac{3}{5}, 0)\).
-
This is a linear function in slope-intercept form.
- This is true, as we have successfully rewritten it in slope-intercept form.
-
This is equivalent to \(y = \frac{5}{6}x + 2\) when converted to slope-intercept form.
- This is false, since we derived it as \(y = \frac{6}{5}x + 2\).
Thus, two statements are incorrect, and only the second statement is true.
Therefore, the correct conclusion is that only one of the above statements is true, not "two of these are correct."