9.04 Simultaneous equations

So far we've only been solving equations with one variable. We know that rearranging $x+2=5$x+2=5 will give us a unique answer of $3$3 for $x$x - but what happens when we have more than one variable in our equation?

Exploration

Consider the equation $x+y=6$x+y=6. This could have the solution $x=2,y=4$x=2,y=4, or it could have the solution of $x=40,y=-34$x=40,y=−34. In fact, there are infinitely many solutions to this equation! Just choose any value for $x$x, and then use the equation to find the corresponding value of $y$y required to make the equation hold true.

If we have two equations with the same two variables in them (often $x$x and $y$y), we call the set of equations system of equations. They are also commonly referred to as simultaneous equations.

We might be interested in finding a common pair of $x$x- and $y$y-values that satisfy both equations simultaneously. If we can find a pair of values for $x$x and $y$y that successfully do this, then we will have found a unique solution to our system.

The graphical method

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

Practice question

Question 1

The substitution method

One way to solve simultaneous equations is called the substitution method. The simultaneous equations we are going to work with usually involve $2$2 variables and $2$2 equations, and this method works by solving for one variable first through 'substituting' one equation into the other.

Consider the equations $3y+4=x$3y+4=x and $y=2+x$y=2+x, for example. We can combine the two equations by substituting $y=2+x$y=2+x into $3y+4=x$3y+4=x. This gives us:

Now that we know the value of $x$x, we can solve for $y$y by using the equation we already have that has $y$y as the subject: $y=2+x$y=2+x. So we have $y=2+\left(-5\right)$y=2+(−5) which is equal to $-3$−3. So our answer is $\left(-5,-3\right)$(−5,−3).

Practice questions

Question 2

Question 3

The elimination method

There's another equally useful method to solve simultaneous equations - the elimination method. 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.

We can eliminate variables from a system of simultaneous equations by adding or subtracting:

For example, we sould add $2x-y=1$2x−y=1 and $5x+y=2$5x+y=2 to get $7x=3$7x=3, which doesn't have any $y$y terms.

As another example, we could subtract $3y-x=2$3y−x=2 from $3y-2x=0$3y−2x=0 to get $-x=-2$−x=−2, which also doesn't have any $y$y terms.

We can see that we should use addition when the coefficients of the variable we want to eliminate are equal and opposite in sign, and subtraction when they're equal and the same sign - but what happens when they don't have the same coefficients at all?

Using multiplication and division

When we don't have the same value coefficients for the variables we want to eliminate, we can multiply or divide the whole equation by a constant until it gets to the coefficient that we want.

For example, consider $x+3y=5$x+3y=5 (equation (1)) and $2y+2x=1$2y+2x=1 (equation (2)). Suppose we want to eliminate the $x$x terms first. Notice that the coefficient of $x$x is $1$1 in equation (1) and $2$2 in equation (2). The idea then is to multiply equation (1) by $2$2 so that the coefficient of $x$x is $2$2 in both equations. Then we can apply the same logic as before and add or subtract the equations. Let's see this in action:

Multiplying equation (1) by $2$2:

Now let's find the difference between equation (3) and equation (2):

Now that we have our value for $y$y it's a simple case of substituting it back in any of the equations to get our value for $x$x. Let's use equation (1):

Practice questions

Question 4

Question 5

Question 6

Solving simultaneous equations using a CAS calculator

A CAS calculator can be utilised in two different ways to solve a linear system of equations. We can use a graphical approach, where we graph both lines on an $xy$xy-plane and use the CAS calculator to find the point of intersection. Or we can use the solve function of the CAS calculator to find the point of intersection immediately with no graphing involved.

Select your brand of calculator below to view instructions for these two approaches.

Casio Classpad

TI Nspire

Practice questions

Question 7

Question 8

Question 9