Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &3 b+2 c=46\\\ &5 c+b=11 \end{aligned}$$

Short answer

The solution to the system of equations is \(b = 16\) and \(c = -1\).

Step-by-step solution

Rewrite the Equations

Arrange both equations in Standard Form (AX + BY = C), so that the terms involving one of the variables line up with each other. In this case, the equations become \(3b + 2c = 46\) and \(b + 5c = 11\).

Find coefficients to Eliminate a Variable

To use linear combinations, need to find a way to add or subtract equations so that one variable cancels out. A good way to do this is to manipulate the equations so that the coefficients of 'b' or 'c' in each equation are negations of each other. In this case, it is possible to manipulate the second equation to cancel out 'b'. This can be done by multiplying the second equation by 3. The equations become: \(3b + 2c = 46\) and \(3b + 15c = 33\).

Combine the Equations

Subtract the second equation from the first one. The 'b' term will be cancelled, and can then solve for 'c'. Doing the subtraction gives: \( -13c=13\). Divide each side by -13 to solve for 'c': \(c= -1\).

Substitute the Solution into Original Equation

Substitute the solution for 'c' into one of the original equations to find 'b'. Substituting into the second equation gives: \( b + 5(-1) = 11\). Then, \( b=16\).