Solving Word Problems with Linear Systems

Read this article and watch the video. The article describes examples in which systems of equations can be used to solve real-world quantities. After you review, complete problems 1 to 4 and check your answers.

Real-World Application: Price of Fruit 

Peter buys two apples and three bananas for $4. Nadia buys four apples and six bananas for $8 from the same store. How much does one banana and one apple costs?

We must write two equations: one for Peter's purchase and one for Nadia's purchase.

Let's say \begin{align*}a\end{align*} is the cost of one apple and \begin{align*}b\end{align*} is the cost of one banana.

  cost of apples cost of bananas total cost
Peter \begin{align*}2a\end{align*} \begin{align*}3b\end{align*} \begin{align*}2a + 3b = 4\end{align*}
Nadia \begin{align*}4a\end{align*} \begin{align*}6b\end{align*} \begin{align*}4a + 6b = 8\end{align*}

The system of equations that describes this problem is:

\begin{align*}2a + 3b &= 4 \\ 4a + 6b &= 8\end{align*}

Let's solve this system by multiplying the first equation by -2 and adding the two equations:

\begin{align*}-2(2a + 3b = 4) \qquad \qquad \quad -4a - 6b = -8\!\\ {\;} \qquad \qquad \qquad \qquad \ \Rightarrow\!\\ \ 4a + 6b = 8 \qquad \qquad \qquad \qquad \underline{\;\;4a + 6b = 8\;\;}\!\\ {\;}\qquad \qquad \qquad \qquad \qquad \qquad \quad \ \ \ 0 + 0 = 0\end{align*}

This statement is always true. This means that the system is dependent.

Looking at the problem again, we can see that we were given exactly the same information in both statements. If Peter buys two apples and three bananas for $4, it makes sense that if Nadia buys twice as many apples (four apples) and twice as many bananas (six bananas) she will pay twice the price ($8). Since the second equation doesn't give us any new information, it doesn't make it possible to find out the price of each fruit.