A woman is making trail mix for her club. At one store, raisins cost \( \$ 3 \) per box anid peanuls cost \( \$ 4 \) per container She has \( \$ 24 \) to spend. The equation that represents buying \( x \) boxes of raisins and y containers of peanuts is \( 3 x+4 y=24 \), What does the \( x \)-ithercept represent in this equation? Find the \( x \)-intercept if she buys 4 boxes of raisins, how many containers of peanuts can she buy? What does the \( x \)-intercept represent in the equation \( 3 x+4 y=24 \) ? A. The number of boxes of raisins that she can buy is equal to the number of containers of peanuts bought. B. The number of containers of peanuts that she can buy if she buys 0 boxes of raisins. C. The number of boxes of raisins that she can buy if she buys 0 containers of peanuts.

Let's break down the problem step by step. ### Step 1: Understanding the Equation The equation given is: \[ 3x + 4y = 24 \] where: - \( x \) is the number of boxes of raisins, - \( y \) is the number of containers of peanuts. ### Step 2: Finding the \( x \)-intercept The \( x \)-intercept occurs when \( y = 0 \). To find the \( x \)-intercept, we can set \( y \) to 0 in the equation and solve for \( x \): \[ 3x + 4(0) = 24 \implies 3x = 24 \implies x = \frac{24}{3} = 8 \] Thus, the \( x \)-intercept is \( 8 \). ### Step 3: Interpretation of the \( x \)-intercept The \( x \)-intercept represents the number of boxes of raisins that can be bought if no containers of peanuts are purchased. Therefore, the correct interpretation is: **C. The number of boxes of raisins that she can buy if she buys 0 containers of peanuts.** ### Step 4: Finding the number of containers of peanuts if she buys 4 boxes of raisins Now, if she buys \( 4 \) boxes of raisins, we can substitute \( x = 4 \) into the original equation to find \( y \): \[ 3(4) + 4y = 24 \implies 12 + 4y = 24 \implies 4y = 24 - 12 \implies 4y = 12 \implies y = \frac{12}{4} = 3 \] Thus, if she buys \( 4 \) boxes of raisins, she can buy \( 3 \) containers of peanuts. ### Summary of Results - The \( x \)-intercept is \( 8 \). - The interpretation of the \( x \)-intercept is **C**. - If she buys \( 4 \) boxes of raisins, she can buy \( 3 \) containers of peanuts.