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Solve the system by the addition method. \( \begin{array}{r}3 x-4 y=-7 \\ 5 x-3 y=-8\end{array} \) Select the correct choice below and fill in any answer boxes present in your choice. A. The solution set is \( \} \). (Simplify your answer. Type an ordered pair.) B. There are infinitely many solutions. C. There is no solution.

Ask by Rowe French. in the United States
Mar 12,2025

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The solution is \( x = -1 \) and \( y = 1 \).

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The Deep Dive

To solve the system of equations using the addition method, we can first manipulate the equations to eliminate one of the variables. Starting with the equations: 1. \(3x - 4y = -7\) (Equation 1) 2. \(5x - 3y = -8\) (Equation 2) To eliminate \(y\), we can multiply Equation 1 by 3 and Equation 2 by 4 to align the coefficients of \(y\): \[ 3(3x - 4y) = 3(-7) \implies 9x - 12y = -21 \] \[ 4(5x - 3y) = 4(-8) \implies 20x - 12y = -32 \] Next, we'll subtract the first modified equation from the second: \[ (20x - 12y) - (9x - 12y) = -32 - (-21) \] This simplifies to: \[ 20x - 9x = -32 + 21 \implies 11x = -11 \] Dividing both sides by 11 gives: \[ x = -1 \] Now that we have \(x\), we can substitute \(x = -1\) into one of the original equations to find \(y\). Let's use Equation 1: \[ 3(-1) - 4y = -7 \] \[ -3 - 4y = -7 \] \[ -4y = -7 + 3 \implies -4y = -4 \] \[ y = 1 \] Thus, the solution to the system of equations is \( (-1, 1) \). This leads us to the conclusion: A. The solution set is \( (-1, 1) \).

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