Questions: Solve the system of equations by the addition method. 12x - 2y = 10 4x + 5y = 9 Select the correct choice below and fill in any answer boxes present in your choice. A. The solution set is 3. (Simplify your answer. Type an ordered pair.) B. There are infinitely many solutions. C. There is no solution.

Solution Steps

To solve the system of equations using the addition method, we need to eliminate one of the variables by adding or subtracting the equations. We can achieve this by multiplying the equations by suitable constants so that the coefficients of one of the variables are opposites. Then, we add the equations to eliminate that variable and solve for the remaining variable. Finally, we substitute back to find the other variable.

We start with the given system of equations: \[ \begin{aligned} 12x - 2y &= 10 \\ 4x + 5y &= 9 \end{aligned} \]

To eliminate one of the variables, we can multiply the second equation by 3 to make the coefficient of \(x\) in both equations the same: \[ \begin{aligned} 12x - 2y &= 10 \\ 12x + 15y &= 27 \end{aligned} \]

Subtract the first equation from the second to eliminate \(x\): \[ (12x + 15y) - (12x - 2y) = 27 - 10 \] \[ 17y = 17 \] \[ y = 1 \]

Substitute \(y = 1\) back into the first equation: \[ 12x - 2(1) = 10 \] \[ 12x - 2 = 10 \] \[ 12x = 12 \] \[ x = 1 \]

Final Answer