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.

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