Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &a-b=8\\\ &a+b=20 \end{aligned}$$

Short answer

The solution of the system of equations is \(a = 14\), \(b = 6\).

Step-by-step solution

Rewrite the equation

First, rewrite the first equation as \(a = b + 8\) for future use.

Substitute the rewritten equation into the other equation

Substitute \(a\) in the second equation with \(b + 8\) from the rewritten first equation, this gives us \(b + 8 + b = 20\).

Solve for the variable

The equation from the previous step simplifies to \(2b + 8 = 20\), Then subtract 8 from both sides which gives us \(2b = 12\), finally by dividing by 2, we get \(b = 6\).

Substitute the variable into the rewritten equation

Now that \(b = 6\), we substitute this into the equation from Step 1, which gives us \(a = 6 + 8 = 14\).

State the solution

The solution to the system of equations is: \(a = 14\) and \(b = 6\).