Question

Use linear combinations to solve the system of linear equations. $$\begin{aligned} &5 e+4 f=9\\\ &4 e+5 f=9 \end{aligned}$$

Short answer

The solution of the system of equations is \(e = 1\) and \(f = 1\).

Step-by-step solution

Multiply both equations by suitable numbers to obtain the same coefficient for f/e

Multiplying the first equation by 4 and the second equation by 5 gives us:\[\begin{aligned} &20 e+16 f=36\ &20 e+25 f=45 \end{aligned}\]

Subtract the second equation from the first

Subtracting these equations:\[20e+16f - (20e+25f) = 36-45\]gives us: \[-9f = -9\]

Solve for variable f

To solve for f, we divide both sides by -9:\[f = 1\]

Substitute f in one of the initial equations and solve for e

Substitute \(f = 1\) in the first equation \(5e+4f=9\) to solve for e:\[5e + 4(1) = 9 \Rightarrow 5e = 5 \Rightarrow e = 1\]