# Solve Simultaneous Equations By Graphing

In 2012, SpaceX successfully launched the world's first privately-owned rocket to the International Space Station
which orbits 220 miles (350 kilometres) above the Earth. **The path of the rocket and the path of the ISS intersected at one point in space**.
The mathematics to do this is called **solving the equations simultaneously**.

Here, we will graph 2 linear equations and identify **the point of intersection at which the 2 graphed lines meet**.
At this point, the equations have the **same x-value and y-value**.

## Example One - Solving Two Linear Equations

Solve the equations y = –x + 5 and y = ^{1}⁄_{2} x + 2

**Answer:**

Point of intersection = ( 2 , 3 )

The simultaneous solution for both equations is x = 2 and y = 3.

## Example Two - Drag Car Overtaking

A fast drag car gives a head start of 2 metres to a slow drag car, but still overtakes it.

If x is the time in seconds and y is the distance travelled in metres,

- the equation for the fast car is y = 4x
- the equation for the slow car is y = 3x + 2

At what point (x,y) or (time, distance) does the fast car overtake the slow car?

**Answer:**

Fast car (blue line in graph)

x | 0 | 1 | 2 | 3 |

y = 4x | 0 | 4 | 8 | 12 |

Slow car (red line in graph)

x | 0 | 1 | 2 | 3 |

y = 3x + 2 | 2 | 5 | 8 | 11 |

The point of intersection is (2,8).

This means that the fast car overtakes the slow car after 2 seconds and at 8 metres from the start.