Question

which of the following conversions is correct?
13x - 11y = 21 to y = \frac{11}{13}x - \frac{21}{13}

y + 9 = \frac{13}{11}(x + 6) to 11x - 13y = 21

-9x - 13y = -11 to y = -\frac{9}{13}x + \frac{11}{13}

y = \frac{13}{11}x - \frac{21}{11} to 13x - 11y = -21

Explanation:

Step1: Analyze Option 1

Start with \(13x - 11y = 21\). Solve for \(y\):
Subtract \(13x\) from both sides: \(-11y = -13x + 21\)
Divide by \(-11\): \(y=\frac{13}{11}x - \frac{21}{11}\). This does not match the given \(y = \frac{11}{13}x - \frac{21}{13}\), so Option 1 is wrong.

Step2: Analyze Option 2

Start with \(y + 9=\frac{13}{11}(x + 6)\). Multiply both sides by 11: \(11(y + 9)=13(x + 6)\)
Expand: \(11y + 99 = 13x + 78\)
Rearrange: \(-13x + 11y = 78 - 99\) → \(-13x + 11y=-21\) → \(13x - 11y = 21\). This does not match \(11x - 13y = 21\), so Option 2 is wrong.

Step3: Analyze Option 3

Start with \(-9x - 13y=-11\). Solve for \(y\):
Add \(9x\) to both sides: \(-13y = 9x - 11\)
Divide by \(-13\): \(y=-\frac{9}{13}x+\frac{11}{13}\). This matches the given conversion, so Option 3 is correct (we can check Option 4 for completeness).

Step4: Analyze Option 4

Start with \(y=\frac{13}{11}x - \frac{21}{11}\). Multiply both sides by 11: \(11y = 13x - 21\)
Rearrange: \(13x - 11y = 21\), not \(13x - 11y=-21\), so Option 4 is wrong.

Answer:

The correct option is the third one: \(-9x - 13y = -11\) to \(y = -\frac{9}{13}x+\frac{11}{13}\) (the option with \(-9x - 13y = -11\) converted to \(y = -\frac{9}{13}x+\frac{11}{13}\))