To solve the problem of finding \( y \) in terms of \( x \) given that the sum of the interior angles of a pentagon is \( 6x + 6y \), we first need to understand the formula for the sum of the interior angles of a polygon. ### Step 1: Understanding the Formula for Interior Angles The formula for the sum of the interior angles of a polygon with \( n \) sides is given by: \[ \text{Sum of interior angles} = (n - 2) \times 180^\circ \] For a pentagon, \( n = 5 \). Therefore, we can substitute \( n \) into the formula: \[ \text{Sum of interior angles} = (5 - 2) \times 180^\circ = 3 \times 180^\circ = 540^\circ \] ### Step 2: Setting Up the Equation According to the problem, the sum of the interior angles is also expressed as \( 6x + 6y \). Therefore, we can set up the equation: \[ 6x + 6y = 540 \] ### Step 3: Simplifying the Equation To simplify this equation, we can divide every term by 6: \[ x + y = 90 \] ### Step 4: Solving for \( y \) Now, we want to express \( y \) in terms of \( x \). We can rearrange the equation: \[ y = 90 - x \] ### Conclusion: Identifying the Correct Option From our calculations, we find that \( y = 90 - x \). This matches option **A**. ### Step 5: Analyzing Other Options Now, let's analyze the other options to understand why they are incorrect: - **Option B: \( y = 150 - x \)** If we substitute \( x = 0 \), then \( y = 150 \). This does not satisfy the equation \( x + y = 90 \) since \( 0 + 150 \neq 90 \). - **Option C: \( y = 60 - x \)** If we substitute \( x = 30 \), then \( y = 30 \). This gives \( 30 + 30 = 60 \), which does not satisfy \( x + y = 90 \). - **Option D: \( y = 120 - x \)** If we substitute \( x = 30 \), then \( y = 90 \). This gives \( 30 + 90 = 120 \), which again does not satisfy \( x + y = 90 \). ### Summary of Key Points - The sum of the interior angles of a pentagon is \( 540^\circ \). - The equation derived from the problem is \( 6x + 6y = 540 \), which simplifies to \( x + y = 90 \). - The correct expression for \( y \) in terms of \( x \) is \( y = 90 - x \), corresponding to option **A**. - Other options do not satisfy the derived equation, confirming their incorrectness. This thorough breakdown should help you understand how to approach similar problems involving the sum of interior angles in polygons.