Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &10 m+16 n=140\\\ &5 m-8 n=60 \end{aligned}$$

Short answer

The solution to the system of equations is \(m = 13\) and \(n = 0.625\).

Step-by-step solution

Multiply Equations

To use linear combinations, it is often useful to manipulate the equations such that addition or subtraction will eliminate one variable. Multiply the second equation by 2 will give \(10m - 16n = 120\). Now we have two equations with same coefficients for \(m\) and \(n\), which can be subtracted to eliminate \(n\).

Subtract the Equations

Subtract the modified second equation from the first equation to eliminate \(n\). This gives \(10m + 16n - (10m - 16n) = 140 - 120, i.e 32n = 20\)

Solve for n

To find the value of \(n\), divide by 32 on both sides, \(n = 20 / 32 = 0.625\). So, we've found the value of \(n\).

Substitute n into the original equation

Now substitute \(n = 0.625\) into one of the original equation, let's use the first one, \(10m + 16*0.625 = 140\), which simplifies to \(10m + 10 = 140\)

Solve for m

Subtract 10 from both sides and then divide by 10 to find the value of \(m\), \(m = (140 - 10) / 10 = 13\). Therefore, we've found the value of \(m\).