Introduction
When we want to solve two equations at the same time, we are looking for the values our variables can take such that both equations will be true. This is referred to as solving simultaneous equations.
Ideas
Equations as graphs
The equations that we will be working with are used to describe the relationship between two variables, usually x and y. As such, we can express these equations graphically as lines on the xy-plane.
If we have two lines on the xy-plane, any point at which those two lines intersect will have an x and y-value satisfying both equations.
The coordinates of the point of intersection of two lines is the solution to the equation of both lines simultaneously. So one way to solve any pair of simultaneous equations is to plot their graphs and find any points of intersection.
If two lines are parallel and are not the same line, they will never intersect. Otherwise, there will be some point of intersection that we can use to solve the equations simultaneously.
Since parallel lines always have the same slope, we can use the gradient of two equations to check whether or not their graphs will have a point of intersection. If their gradients are the same then there will be no values for x and y that solve the equations simultaneously.
Examples
Example 1
Consider the following linear equations:
\begin{aligned} y&=5x-7 \\ y &=-x+5 \end{aligned}
Sketch the two lines representing these equations on a cartesian plane.
State the values of x and y which satisfy both equations.
We can solve any pair of simultaneous equations by plotting their graphs and finding the point of intersection.
If two different lines have the same gradient, then they will be parallel and will have no point of intersection. So their equations will have no simultaneous solution.