\begin{tabular}{l} Do Now: \\ Solve the problem below using BOTH elimination and substitution \\ ㅍmination Method \\ \( \qquad \)\begin{tabular}{l|l} \\ \( \qquad y x+3 y=16 \) & Substitution Method \\ \( 3 x-5 y=-4 \) & \( 3 x+3 y=16 \) \\ & \( 3 x-5 y=-4 \)\end{tabular} \\ \hline\end{tabular}

To solve the system of equations using the elimination method, we can first write the equations in a more organized way: 1. \(3x + 3y = 16\) (Equation A) 2. \(3x - 5y = -4\) (Equation B) Next, we can eliminate \(x\) by manipulating the equations. We can multiply Equation A by 5 and Equation B by 3 to align the \(y\) coefficients: - Multiply Equation A by 5: \(15x + 15y = 80\) - Multiply Equation B by 3: \(9x - 15y = -12\) Now, we have: 1. \(15x + 15y = 80\) 2. \(9x - 15y = -12\) Next, we add them: \[ (15x + 15y) + (9x - 15y) = 80 - 12 \] This simplifies to: \[ 24x = 68 \] Dividing both sides by 24 gives: \[ x = \frac{17}{6} \] Now substitute \(x\) back into one of the original equations to find \(y\). Using Equation A: \[ 3\left(\frac{17}{6}\right) + 3y = 16 \] This simplifies to: \[ \frac{51}{6} + 3y = 16 \] To clear the fraction, multiply everything by 6: \[ 51 + 18y = 96 \] Subtracting 51 from both sides results in: \[ 18y = 45 \] Dividing both sides by 18 yields: \[ y = \frac{5}{2} \] Thus, the solution using the elimination method is \(x = \frac{17}{6}\) and \(y = \frac{5}{2}\). Now, for the substitution method, we start from the original equations and isolate one variable. Let's isolate \(y\) from Equation A: Starting with \(3x + 3y = 16\): \[ 3y = 16 - 3x \] \[ y = \frac{16 - 3x}{3} \] Now, substitute \(y\) into Equation B: \[ 3x - 5\left(\frac{16 - 3x}{3}\right) = -4 \] Distributing gives: \[ 3x - \frac{80 - 15x}{3} = -4 \] Clearing the fraction by multiplying by 3 results in: \[ 9x - (80 - 15x) = -12 \] Simplifying this gives: \[ 9x - 80 + 15x = -12 \] \[ 24x - 80 = -12 \] Adding 80 to both sides yields: \[ 24x = 68 \] Thus, \(x = \frac{17}{6}\). We can find \(y\) again using our previous \(y\) expression: \[ y = \frac{16 - 3(\frac{17}{6})}{3} \] Calculating gives: \[ y = \frac{16 - \frac{51}{6}}{3} = \frac{\frac{96 - 51}{6}}{3} = \frac{\frac{45}{6}}{3} = \frac{15}{6} = \frac{5}{2} \] Therefore, we confirm that the solution is \(x = \frac{17}{6}\) and \(y = \frac{5}{2}\) using both methods. Happy solving! 🎉