Example 5.3.2

Alicia has \(\$1000\) to give to her two grandchildren for New Year's. She would like to give the older grandchild \(\$120\) more than the younger grandchild, because that is the cost of the older grandchild's college textbooks this term. How much money should she give to each grandchild?

To answer this question, we will demonstrate a new technique. You may have a very good way for finding how much money Alicia should give to each grandchild, but right now we will try to see this new method.

Let \(A\) be the dollar amount she gives to her older grandchild, and \(B\) be the dollar amount she gives to her younger grandchild. (As always, we start solving a word problem like this by defining the variables, including their units.) Since the total she has to give is \(\$1000\text{,}\) we can say that \(A+B=1000\text{.}\) And since she wants to give \(\$120\) more to the older grandchild, we can say that \(A-B=120\text{.}\) So we have the system of equations:

\begin{align*} \left\{ \begin{aligned} A+B \amp = 1000 \\ A-B \amp = 120 \end{aligned} \right. \end{align*}

We could solve this system by substitution as we learned in Section 5.2, but there is an easier method. If we add together the left sides from the two equations, it should equal the sum of the right sides:

\begin{align*} \begin{aligned}A+B\\{}+A-B\end{aligned}\amp=\begin{aligned}1000\\{}+120\end{aligned}\\ \end{align*}

So we have:

\begin{align*} 2A\amp=1120 \end{align*}

Note that the variable \(B\) is eliminated. This happened because the “\({}+B\)” and the “\({}-B\)” were perfectly in shape to cancel each other out. With only one variable left, it doesn't take much to finish:

\begin{align*} 2A\amp=1120\\ A\amp=560 \end{align*}

To finish solving this system of equations, we need the value of \(B\text{.}\) For now, an easy way to find \(B\) is to substitute in our value of \(A\) into one of the original equations:

\begin{align*} A+B\amp=1000\\ \substitute{560}+B\amp=1000\\ B\amp=440 \end{align*}

To check our work, substitute \(A=560\) and \(B=440\) into the original equations:

\begin{align*} A+B\amp=1000\amp A-B\amp=120\\ \substitute{560}+\substitute{440}\amp\stackrel{?}{=}1000\amp \substitute{560}-\substitute{440}\amp\stackrel{?}{=}120\\ 1000\amp\stackrel{\checkmark}{=}1000\amp 120\amp\stackrel{\checkmark}{=}120 \end{align*}

This confirms that our solution is correct. In summary, Alicia should give \(\$560\) to her older grandchild, and \(\$440\) to her younger grandchild.