Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned}&2 m-4=4 n\\\&m-2=n\end{aligned}$$

Short answer

The solution to the system of equations is \(m = 4\) and \(n = 2\)

Step-by-step solution

Isolate a variable in the second equation

We can isolate n in the second equation \(m-2=n\) by rewriting it as \(n = m - 2\)

Substitute \(n = m - 2\) in the first equation

Substitute \(n = m - 2\) in the first equation \(2m - 4 = 4n\) to get a linear equation with only m: It now becomes \(2m - 4 = 4(m - 2)\)

Solve for m in the linear equation

Solving the equation \(2m - 4 = 4(m - 2)\) simplifies to \(2m = 4m - 8\), which simplifies further to \(2m = 8\). Finally, we see that \(m = 4\)

Substitute m into the original equation to find n

Now that we have found the value of m, we substitute it into the equation \(n = m - 2\) which we rearranged at first. This yields \(n = 4 - 2 = 2\)