Question

Use linear combinations to solve the system. (Review 7.3 ) $$\begin{aligned}&10 x-3 y=17\\\&-7 x+y=9\end{aligned}$$

Short answer

The solution to the system of equations is \(x = -4\) and \(y = -19\)

Step-by-step solution

Multiply the equations

For the given system of equations \(10x - 3y = 17\) and \(-7x + y = 9\), it will be effective to multiply the second equation by 3, so that the coefficients for 'y' in both equations will cancel each other out when added. After multiplying the second equation by 3, the new equations to be solved are \(10x - 3y = 17\) and \(-21x + 3y = 27\)

Add the equations

The aim now is to add the two equations from Step 1, such that the coefficients of 'y' will cancel out each other. This results in \(-11x = 44\)

Solve for variable x

From Step 2, the variable x can be found by dividing the equation \(-11x = 44\) by -11. This gives \(x = -4\)

Substitute x into second equation

Substitute this value of x into the second equation (\(-7x + y = 9\)), we get: \(-7(-4) + y = 9\), from which we find that \(y = -19\) after simplifying

Verification

To verify these solutions, \(x = -4\) and \(y = -19\), should satisfy both the original equations \(10x - 3y = 17\) and \(-7x + y = 9\). By substituting these values into both equations, it is confirmed that they are solutions.