Question

Use linear combinations to solve the system. (Review 7.3 ) $$\begin{aligned}&8 x-5 y=100\\\&2 x+\frac{1}{2} y=4\end{aligned}$$

Short answer

The solution to the system is \(x = 45\) and \(y = 68\).

Step-by-step solution

Multiply Both Equations to Match Coefficients

Multiply equation 1 by \(2\) and equation 2 by \(16\). This will make the coefficients of \(x\) in both equations the same (16), but with opposite signs. The new system of equations becomes: \[16x - 10y = 200\] and \[16x + 8y = 64\] respectively.

Add equations

Add these two equations to cancel out \(x\). That results in: \[-2y = -136\]

Solve for \(y\)

Divide each side of the equation by -2 to solve for \(y\). Thus \(y = 68\)

Substitute \(y = 68\) into first equation

Substitute \(y = 68\) into the original equation: \(8x - 5(68) = 100\)

Solve for \(x\)

Solving the equation for \(x\) gives \(x = 45\)

Write Down The Solution

The solution is a set of points \((45, 68)\), where \(x = 45\) and \(y = 68\). This means that both equations are true when \(x = 45\) and \(y = 68\)