QRSP is a rectangle. Find the length of QA given that QA = 4x - 7 and QS = 5x + 7.

Steps to Solve

1. Set the equations equal Since QA and QS represent the same dimension in rectangle QRSP, we set their expressions equal to each other:

$$ 4x - 7 = 5x + 7 $$

2. Isolate the variable ( x ) We will get all terms involving ( x ) on one side and constant terms on the other. First, subtract ( 4x ) from both sides:

$$ -7 = x + 7 $$

Next, subtract 7 from both sides:

$$ -14 = x $$

3. Substitute ( x ) back into the expression for QA Now that we have the value of ( x ), substitute it back into the expression for QA:

$$ QA = 4(-14) - 7 $$

4. Calculate ( QA ) Now simplify the equation:

$$ QA = -56 - 7 = -63 $$

Since lengths cannot be negative, let's check for errors in the calculations. We should ensure the consistency of the rectangle's dimension.

However, it seems there may have been a sign issue in the calculations. Let's work backward correctly to verify QA:

1. Substitute ( -14 ) into one of the original expressions again, ensuring correctness with ( QS ) as:

$$ QS = 5(-14) + 7 = -70 + 7 = -63 $$

So, QA should be effectively broken down or checked for dimensional properties in context.

To correct: It's noted that lengths must be corrected and integers preserved.