# LINR 3 | Lesson 1 | Explore 2 Solution

One student found from analyzing the tables and graphs that (6.6, 8.3) was the approximate point of the tie.

Check this solution by substituting into each of the equations found:

Blue line:

$$y=\Large \frac{1}{2}\normalsize x+5$$… $$8.3=8.3$$ 😁

Red line:

$$y=\Large \frac{5}{4}\normalsize x$$… $$8.3\neq8.25$$ 😓

So, that guess was a little off.

We can use our equations to find an exact point of intersection.  There are a number of ways to do so.  Below is the substitution method.

##### Substitution Method:

Solve:

$y\,_{\text{blue}}=y\,_{\text{red}}$

$\frac{1}{2}x+5=\frac{5}{4}x$

$5=\frac{3}{4}x$

$x=\frac{20}{3}$ number of minutes where the race is tied.

Then $$y=\Large \frac{1}{2}\cdot \frac{20}{3}\normalsize+5$$

$$y=\Large \frac{25}{3}$$ feet at which the race is tied.

Verify in other equation… 😁 $\frac{25}{3}=\,? \frac{5}{4}\cdot\frac{20}{3}$

I.e. The race is tied at $$\Large \frac{20}{3}$$ minutes, when the insects are $$\Large \frac{25}{3}$$ feet from the start.

The Arnie and Sally race represents a linear system. The point $$\left( \Large \frac{20}{3}\normalsize, \Large \frac{25}{3}\right)$$, where they tie is the solution to the system.

A system of linear equations consists of two or more equations considered together. A solution to the system is an (x,y) pair that satisfies all of the equations. A solution represented in a graph is a point where all of the lines intersect, in a table it is where the coordinates in each table are the same.