Solve the system by the addition method. \[ \begin{array}{l}5 x-2 y=11 \\ 4 x+3 y=18\end{array} \]
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To solve the system by the addition method, we'll first manipulate the equations so we can eliminate one variable when we add the two equations together. The given equations are: 1) \( 5x - 2y = 11 \) 2) \( 4x + 3y = 18 \) First, we will multiply each equation by a coefficient such that the \(y\) coefficients are opposites. We can multiply the first equation by 3 and the second one by 2: 1) \( 3(5x - 2y) = 3(11) \) This simplifies to: \( 15x - 6y = 33 \) (Equation 3) 2) \( 2(4x + 3y) = 2(18) \) This simplifies to: \( 8x + 6y = 36 \) (Equation 4) Now we have a new system of equations: 3) \( 15x - 6y = 33 \) 4) \( 8x + 6y = 36 \) Next, we can add Equation 3 and Equation 4 to eliminate \(y\): \[ (15x - 6y) + (8x + 6y) = 33 + 36 \] This simplifies to: \[ 23x = 69 \] Now we can solve for \(x\): \[ x = \frac{69}{23} = 3 \] Now that we have the value of \(x\), we can substitute it back into one of the original equations to find \(y\). We'll use Equation 1: \[ 5(3) - 2y = 11 \] \[ 15 - 2y = 11 \] \[ -2y = 11 - 15 \] \[ -2y = -4 \] \[ y = \frac{-4}{-2} = 2 \] The solution to the system of equations is: \[ (x, y) = (3, 2) \]
