SYSTEMS OF EQUATIONS in TWO VARIABLES

A system of equations is a collection of two or more equations with the same set of unknowns. In solving a system of equations, we try to find values for each of the unknowns that will satisfy every equation in the system.

The equations in the system can be linear or non-linear. This tutorial reviews systems of linear equations.

A problem can be expressed in narrative form or the problem can be expressed in algebraic form.

Let's start with an example stated in narrative form. We'll convert it to an equivalent equation in algebraic form, and then we will solve it.

Example 6:

Two planes start from the same airport and fly in different directions. The second plane starts one-half hour after the first plane, but its speed is 50 miles per hour faster. Find the ground speed of each plane if 2 hours after the first plane starts the planes are 2000 miles apart.[Ground speed is the speed of the plane discounting any wind.]

The distance the first plane traveled plus the difference the second plane traveled totals 2000 miles. If we let the phrase ''distance the first plane traveled'' be represented by the symbol $d_{1},$ and the phrase ''distance the second plane traveled'' be represented by the symbol $d_{2}.\bigskip \bigskip$ The sentence ''The distance the first plane traveled plus the difference the second plane traveled totals 2000 miles.'' can be written as

$$\begin{eqnarray*} d_{1}+d_{2} &=&2000 \\ && \\ && \end{eqnarray*}$$

As you know, distance equals the rate $\times$ time. We have two unknowns the speed of the first plane and the speed of the second plane. Let '' the speed of the first plane'' be represented by the symbol $r_{1}$ and let ''the speed of the second plane'' be represented by the symbol $r_{2}.$

The distance the first plane traveled can now be represented by the equation

$$\begin{eqnarray*} && \\ d_{1} &=&2r_{1} \\ && \\ && \end{eqnarray*}$$

The distance the second plane traveled can now be represented by the equation

$$\begin{eqnarray*} && \\ d_{2} &=&1.5r_{2} \\ && \\ && \end{eqnarray*}$$

The phrase ''the speed of the second plane is 50 miles per hour faster than the first plane can be represented by the equation

$$\begin{eqnarray*} && \\ r_{2} &=&r_{1}+50 \\ && \\ && \end{eqnarray*}$$

So now you can write the distance equation for plane 2 as

$$\begin{eqnarray*} && \\ d_{2} &=&1.5\left( r_{1}+50\right) \\ && \\ d_{2} &=&1.5r_{1}+75 \\ && \\ && \end{eqnarray*}$$

We know that the sum of the distances of the planes from the starting point is 2000.

$$\begin{eqnarray*} && \\ && \\ d_{1}+d_{2} &=&2000 \\ && \\ d_{1}+d_{2} ... ... && \\ r_{1} &=&550 \\ && \\ r_{2} &=&600 \\ && \\ && \end{eqnarray*}$$

Let's check our answers.

First Plane: speed $\times$ time = $550\times 2=1100\bigskip$ Second Plane: speed $\times$ time = $600\times 1.5=900\bigskip$ The sum of the distances is $2000.\bigskip\bigskip\bigskip$

If you would like to test yourself by working some problem similar to this example, click on Problem.

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Author: Nancy Marcus

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