We use simultaneous equations to solve for values of variables that make two or more equations simultaneously true. This has many applications from finance to computing. We will limit our studies to two equations in two unknowns.
We have two algebraic methods to solve simultaneous equations - elimination or substitution. Ultimately we are trying to combine our equations in a way that reduces them down to only one unknown.
The substitution method
As the simultaneous equations we usually work with involve two variables and two equations, the substitution method works by solving one variable first through 'substituting' one equation into the other. It is also vital when dealing with systems of equations involving non-linear relationships as we will see in later examples.
Worked example
Example 1
Solve the pair of equations $x=3y+4$x=3y+4 and $2+x-y=0$2+x−y=0 simultaneously.
Think: As we already have one equation with $x$x as the subject, we can just substitute this into the other equation.
Do: To begin, we substitute the value of $x$x from the first equation into the second giving:
| $2+3y+4-y$2+3y+4−y | $=$= | $0$0 |
| $2y$2y | $=$= | $-6$−6 |
| $y$y | $=$= | $-3$−3 |
We can use either of the original equations to find the matching value of $x$x as follows:
| $x$x | $=$= | $3y+4$3y+4 |
| $x$x | $=$= | $3\times-3+4$3×−3+4 |
| $x$x | $=$= | $-5$−5 |
Hence, the solution to these two simultaneous equations is $x=-5$x=−5 and $y=-3$y=−3. These are the values of $x$x and $y$y that make both of these equations true simultaneously.
The elimination method
The elimination method is equally useful when working with systems of linear simultaneous equations. It works by adding or subtracting equations from one another to eliminate one variable, leaving us with the other variable to solve on its own.
Worked example
Example 2
Solve $2x-y=1$2x−y=1 and $5x+y=2$5x+y=2 simultaneously using the elimination method.
Think: It is useful to look if the coefficients of one of the variables are the same. We can then do a straight addition or subtraction of the equations to eliminate this variable. If no variables have the same coefficients, we have to make them equal for one variable by multiplying each equation appropriately. For this pair of equations the coefficients of the $y$y terms will allow us to eliminate this variable by direct addition of equations.
Do: The first step is to number the equations as follows:
| $2x-y$2x−y | $=$= | $1$1 | equation (1) | |
| $5x+y$5x+y | $=$= | $2$2 | equation (2) |
Adding equation (2) to equation (1) we get:
| $2x+5x-y+y$2x+5x−y+y | $=$= | $1+2$1+2 |
| $7x$7x | $=$= | $3$3 |
| $x$x | $=$= | $\frac{3}{7}$37 |
We can use either of the original equations to find the matching value of $y$y as follows:
| $2x-y$2x−y | $=$= | $1$1 |
| $2\times\frac{3}{7}-y$2×37−y | $=$= | $1$1 |
| $y$y | $=$= | $\frac{6}{7}-1$67−1 |
| $y$y | $=$= | $-\frac{1}{7}$−17 |
Hence, the solution to these two simultaneous equations is $x=\frac{3}{7}$x=37 and $y=-\frac{1}{7}$y=−17. These are the values of $x$x and $y$y that make both of these equations true simultaneously.
- Name the equations by writing (1) and (2) next to them, and whenever we create new equations out of one or both of them, we can name them (3), (4), etc. When adding two equations we can write it in shorthand as (3) + (1), and similarly for all the other operations. This helps to keep our ideas in order and not get confused.
- When dealing with equations with big numbers, see if they can be simplified before beginning to solve them. For example, $2x-4=6y$2x−4=6y can be simplified to $x-2=3y$x−2=3y without changing the values of $x$x and $y$y.
- Remember to check your answers by substituting your $x$x and $y$y values back into the original two equations.
Remember to solve for the values of both $x$x and $y$y! Check if you have both at the end of every simultaneous question problem unless told otherwise.
Practice questions
Question 1
We want to solve the following system of equations using the substitution method.
| Equation 1 | $y=5x+34$y=5x+34 |
| Equation 2 | $y=3x+18$y=3x+18 |
First solve for $x$x.
Hence, solve for $y$y.
Question 2
Use the elimination method to solve for $x$x and $y$y.
| Equation 1 | $-6x-2y=46$−6x−2y=46 |
| Equation 2 | $-30x-6y=246$−30x−6y=246 |
First solve for $x$x.
Now solve for $y$y.
Simultaneous equations involving quadratic functions
Having revised our algebraic techniques for simultaneous equations, we will now think about how this connects with our graphical understanding of functions. As with all algebraic expressions, simultaneous equations can also be shown as graphs on a number plane. Each graph of a function represents all of the possible solutions of that equation. So, when we are finding a solution that solves both of our simultaneous equations, we are looking for the point of intersection of the two graphs.
When we have two linear equations (or straight lines), they will only have one point of intersection. However, when we have a quadratic (parabola) and a linear equation (straight line), there can be:
- two points of intersection
- one point of intersection
- or even no points of intersection (meaning there are no solutions)
If we were to solve two quadratic equations simultaneously, we could similarly have $0$0, $1$1 or $2$2 solutions. Can you visualise these three scenarios too?
Some of the trickiest simultaneous equations questions require you to move back and forth between a graphical understanding and the real life context presented. Make sure you carefully read the question to see if the answer should be presented as a set of coordinates of the point of intersection of the graphs, or in terms of actual units of measurement related to the question!
Practice questions
Question 3
Solve the following equations simultaneously:
| Equation 1 | $y=2x^2+14x$y=2x2+14x |
| Equation 2 | $y=16x$y=16x |
First solve for $x$x.
For $x=0$x=0, find $y$y.
For $x=1$x=1, find $y$y.
Question 4
Consider the quadratic function $y=x^2+4$y=x2+4 and the linear function $x+y=4$x+y=4.
Graph the quadratic function $y=x^2+4$y=x2+4.
Loading Graph...Graph the line $x+y=4$x+y=4 on the same set of axes.
Loading Graph...How many solutions are there to the equation $x^2+4=-x+4$x2+4=−x+4?
Determine the points of intersection of $y=x^2+4$y=x2+4 and $x+y=4$x+y=4. Write the coordinates on the same line separated by a comma.
Question 5
Consider the graphs of $y=x^2$y=x2 and $y=-3x$y=−3x.
How many points of intersection are there between the two graphs?
One point of intersection occurs at $\left(0,0\right)$(0,0). State the coordinates of the other point of intersection in the form $\left(x,y\right)$(x,y).
Consider the equation $x^2=-3x$x2=−3x. State the solutions of this equation, writing both values on the same line separated by a comma.