Find the solution to the system by the addition (elimination) method. Check your answers. \( \begin{array}{ll}2 s+5 t=12 & \text { (1) } \\ 7 s-10 t=31 & \text { (2) }\end{array} \) What is the solution to the system?

To solve the system of equations using the **elimination (addition) method**, follow these steps: Given the system: \[ \begin{cases} 2s + 5t = 12 \quad \text{(1)} \\ 7s - 10t = 31 \quad \text{(2)} \end{cases} \] **Step 1: Align the equations for elimination.** We aim to eliminate one of the variables by making their coefficients equal in magnitude. Let's eliminate \( t \) by finding the least common multiple (LCM) of the coefficients of \( t \) in both equations. - The coefficients of \( t \) are 5 and -10. - The LCM of 5 and 10 is 10. **Step 2: Multiply the equations to make the coefficients of \( t \) equal.** - Multiply equation (1) by 2 to make the coefficient of \( t \) equal to 10: \[ 2(2s + 5t) = 2 \times 12 \implies 4s + 10t = 24 \quad \text{(3)} \] - Keep equation (2) as is: \[ 7s - 10t = 31 \quad \text{(2)} \] **Step 3: Add equations (3) and (2) to eliminate \( t \).** \[ (4s + 10t) + (7s - 10t) = 24 + 31 \\ 11s = 55 \\ s = \frac{55}{11} = 5 \] **Step 4: Substitute \( s = 5 \) back into one of the original equations to solve for \( t \).** Using equation (1): \[ 2(5) + 5t = 12 \\ 10 + 5t = 12 \\ 5t = 2 \\ t = \frac{2}{5} \] **Step 5: Check the solution in the second equation.** \[ 7(5) - 10\left(\frac{2}{5}\right) = 35 - 4 = 31 \] This satisfies equation (2), confirming that the solution is correct. **Final Solution:** \[ \boxed{\left( s, t \right) = \left( 5,\ \dfrac{2}{5} \right)} \]