###### Example103

The sum of \(4\) times a larger integer and \(5\) times a smaller integer is \(7\text{.}\) When twice the smaller integer is subtracted from \(3\) times the larger, the result is \(11\text{.}\) Find the integers.

Begin by assigning variables to the larger and smaller integer.

Let \(x\) represent the larger integer, and let \(y\) represent the smaller integer.

When using two variables, we need to set up two equations. The first sentence describes a sum and the second sentence describes a difference. The phrase "\(4\) times a larger integer" translates to the expression \(4x\text{,}\) and "\(5\) times a smaller integer" translates to \(5y\text{.}\) Altogether, we get \(4x+5y=7\) for the first equation. Likewise, we get \(3x-2y=11\) for the second equation. This leads to the following system:

We will solve this using the elimination method. To eliminate the variable \(y\) multiply the first equation by \(2\) and the second by \(5\text{.}\)

Next we add the equations in the equivalent system and solve for \(x\text{.}\)

This gives \(x=\frac{69}{23}=3\text{.}\) Now we use this for a substitution into an original equation to find \(y\text{.}\)

The larger integer is \(3\) and the smaller integer is \(-1\text{.}\)