1.01 Introduction to simultaneous equations

We know how to solve equations with one variable. For example, to solve $x+2=5$x+2=5 we can rearrange the equation and get a unique solution of $x=3$x=3, but how do we solve an equation that has more than one variable?

Exploration

Consider $x+y=6$x+y=6. This equation has many solutions, such as the pairs $x=2$x=2, $y=4$y=4 and $x=40$x=40, $y=-34$y=−34. In fact, there are infinite number of solutions to this equation. To find a solution, just pick any value for $x$x, and then ask 'What number adds with $x$x to make $6$6?' This gives the corresponding value of $y$y for that pair.

If we have two equations with the same two variables ($x$x and $y$y), then we call them a system of equations. They are also commonly referred to as simultaneous equations. Here are a pair of simultaneous equations:

$y=x-1$y=x−1

$y=5x-13$y=5x−13

We might be interested in finding a common pair of $x$x- and $y$y-values that satisfies both of these equations simultaneously - and whether or not this is the unique solution. To answer this question we will use a graph of each line.

Graphical representations

On a coordinate plane, the solution of two simultaneous equations is represented by the point of intersection of the graphs of the equations, where the two lines cross. So the $x$x- and $y$y-values of our solution will take the form of coordinates of the intersection point $\left(x,y\right)$(x,y).

Here are the graphs of $y=x-1$y=x−1 and $y=5x-13$y=5x−13.

The point lies on both lines, so it satisfies both equations

We can see that the lines are not parallel and there is one point of intersection. The coordinates of the point are $\left(3,2\right)$(3,2), so the solution to the equations $y=x-1$y=x−1 and $y=5x-13$y=5x−13 is $x=3$x=3 and $y=2$y=2.

We can check this is the correct solution by substituting $x=3$x=3, $y=2$y=2 into the two equations in the system:

Both of these equations are true, so we have found our solution!

In this case, there is exactly one unique solution. With this in mind, will this be true in all cases? How many solutions should be possible for each set of two simultaneous linear equations?

Parallel lines

Parallel lines do not intersect, so how do we solve simultaneous equations that produce parallel lines when graphed?

If a pair of parallel lines don’t have the same graph, then there will be zero points of intersection between them. For example, the graphs of $x+y=0$x+y=0 and $x+y=1$x+y=1 never meet.

This means that no corresponding $x$x- and $y$y-values satisfy both equations simultaneously. In this case, there is no solution.

If a pair of parallels lines do have the same graph, then every point on the line is a point of intersection. For example, if the graphs of $x+y=5$x+y=5 and $2x+2y=10$2x+2y=10 were placed on the same set of axes, we would end up with two lines lying perfectly on top of one another.  

In this case there are infinite solutions.

Practice questions

Question 1

Question 2

Question 3