Explainer: Solving Systems of Linear Equations Using Elimination

In this explainer, we will learn how to solve systems of linear equations using elimination.

When we are asked to solve a system of equations (sometimes called simultaneous equations), we are trying to find any points at which the equations are equal, that is, any points where they intersect. Before looking at how we can solve these questions algebraically, let us look at a graphical example. If we draw the graphs of the equations 𝑦𝑥=1 and 𝑦+𝑥=5, we get the following.

From this, we can see that the graphs intersect at the point (2,3). This is the point, therefore, which is the solution to the system of equations 𝑦𝑥=1 and 𝑦+𝑥=5. We can check that this is correct by substituting the coordinates into each of the equations. For the first, we get (3)(2)=1,

which is correct, and for the second equation we get (3)+(2)=5,

which is also correct.

Further to solving the system graphically, we can take an algebraic approach. There are two main methods for doing this: one is called substitution and the other elimination. Here, we are going to focus on elimination. The method is titled as such as we solve the system of equations by eliminating a variable by adding or subtracting multiples of one or both equations. Often, when solving systems of two equations, elimination is a more desirable method to use when you want to avoid having to rearrange one or both equations. If we consider again the system of equations 𝑦𝑥=1,𝑦+𝑥=5,

we can eliminate the 𝑦 variable by subtracting one equation from the other or eliminate the 𝑥 variable by adding the two equations. Let us look at both methods. If we subtract the equations, we get 𝑦𝑦=0, 𝑥𝑥=2𝑥, and 15=4, giving us 2𝑥=4.

If we then divide through by 2, we find that 𝑥=2.

Remember to be very careful with your negative signs throughout your calculations. At the point we can then substitute 𝑥=2 into either one of our equations to find 𝑦. Here, let us substitute into the first equation: 𝑦(2)=1𝑦=3.

As we can see, these solutions are consistent with what we saw from the graph. Equally, we could have added our two equations which would have given us 𝑦+𝑦=2𝑦, 𝑥+𝑥=0, and 1+5=6, giving 2𝑦=6.

Dividing through by two gives us 𝑦=3.

Again, at this point, we could substitute into either equation to find that 𝑥=2. Generally, to eliminate a variable where both signs are the same we subtract one equation from the other and if we want to eliminate a variable where the signs are different we add the two equations. We may also have to multiply the equations through by a number first to make the value of the two coefficients the same. We will look at examples of each of these cases now.

Example 1: Solving Systems of Equations Using Elimination by Adding

Solve the simultaneous equations 𝑥𝑦=4 and 𝑥+𝑦=14.

Answer

To solve this system of equations, we can start by either adding or subtracting the two equations as the coefficient of each variable is the same. Here, we will add the two equations to eliminate the variable 𝑦: 𝑥𝑦=4+𝑥+𝑦=142𝑥=18.

If we then divide through by 2, we find that 𝑥=9.

We then need to substitute into either one of the two equations. Here, we will substitute into the second as this contains no negatives: 9+𝑦=14.

Then, we subtract 9 from each side to find that 𝑦=5.

Our solutions are, therefore, 𝑥=9 and 𝑦=5, which corresponds to the point (9,5).

Now, let us look at an example where one of the equations needs to be multiplied before we add or subtract the equations to eliminate a variable.

Example 2: Solving Systems of Equations Using Elimination by Subtracting

Solve the simultaneous equations 𝑥+4𝑦=17 and 2𝑥+7𝑦=5.

Answer

Our first step is to identify which of the variables we want to eliminate. Here, the coefficients of the 𝑥 variables are 1 and 2, whereas the coefficients of the 𝑦 variables are 4 and 7. As 2 is a multiple of 1, this means that we can make the coefficients of 𝑥 the same by only multiplying one of the equations. If we multiply the first equation through by 2, we get 2𝑥+8𝑦=34.

We can then subtract the second equation from this equation to eliminate the variable 𝑥: 2𝑥+8𝑦=342𝑥+7𝑦=5𝑦=29.

If we then substitute this value of 𝑦 into either of the original equations, we can calculate the value of 𝑥. If we substitute into the first equation, we get 𝑥+4(29)=17.

Simplifying, we get 𝑥+116=17,

and subtracting 116 from both sides we find that 𝑥=99. The solution to this system of equations is, therefore, 𝑥=99 and 𝑦=29, which corresponds to the point (99,29).

In some instances, to solve an equation by elimination, you will need to multiply both equations through by a number to make the coefficients of one of the variables the same. Let us look at a couple of examples of this now.

Example 3: Solving Systems of Equations Using Elimination Where Each Equation Needs to Be Multiplied

Use the elimination method to solve the given simultaneous equations: 4𝑥2𝑦=4,5𝑥+3𝑦=16.

Answer

With this question, we need to multiply both equations to make the coefficients of either the 𝑥 or the 𝑦 variables the same. Here, we are going to make the coefficients of 𝑦 equal (simply because the numbers have smaller magnitude) but we could equally make the coefficients of 𝑥 equal. If we multiply the first equation through by 3 and the second equation through by 2, we get 12𝑥6𝑦=12,10𝑥+6𝑦=32.

We now have two equations with two coefficients of equal magnitude, but opposite signs. To eliminate the 𝑦 variable, we need to add the two equations to get 22𝑥=44.

If we then divide through by 22, we get 𝑥=2.

In order to find 𝑦, we need to substitute our value of 𝑥 into one of the original equations. We will substitute into the second equation as this contains all positive numbers. We get 5(2)+3𝑦=16,

which simplifies to 10+3𝑦=16.

If we subtract 10 from each side and then divide through by 4, we find that 𝑦=2.

The solution to this system of equations is, therefore, 𝑥=2 and 𝑦=2, which corresponds to the point (2,2).

Example 4: Solving Systems of Equations Using Elimination Where Each Equation Needs to Be Multiplied

Using elimination, solve the simultaneous equations: 4𝑥+6𝑦=40,3𝑥+7𝑦=40.

Answer

With this question, we need to multiply both equations to make the coefficients of either the 𝑥 or the 𝑦 variables the same. Here, we are going to make the coefficients of 𝑥 equal (simply because the numbers have smaller magnitude) but we could equally make the coefficients of 𝑦 equal. If we multiply the first equation through by 3 and the second equation through by 4, we get 12𝑥+18𝑦=120,12𝑥+28𝑦=160.

We now have two equations with two coefficients of equal magnitude. To eliminate the 𝑥 variable, we need to subtract one equation from the other. Here, we will subtract the first from the second to avoid having to work with negative numbers to get 10𝑦=40.

If we then divide through by 10, we get 𝑦=4.

In order to find 𝑥, we need to substitute our value of 𝑦 into one of the original equations. We will substitute into the first equation to get 4𝑥+6(4)=40, which simplifies to 4𝑥+24=40.

If we subtract 24 from each side and then divide through by 4, we find that 𝑥=4.

The solution to this system of equations is, therefore, 𝑥=4 and 𝑦=4, which corresponds to the point (4,4).

On occasion, you will need to rearrange one (or both) of the equations before eliminating a variable. Let us look at an example of this before finishing.

Example 5: Solving a System of Equations by Rearranging and Then Using Elimination

Solve the simultaneous equations 𝑥𝑦=8 and 3𝑥5𝑦+10=0.

Answer

With this question, we can start by rearranging the second equation so it is the same form as the first equation. If we subtract 10 from both sides of the second equation, our system becomes 𝑥𝑦=8,3𝑥5𝑦=10.

We now need to multiply one of the equations to make the coefficients of either 𝑥 or 𝑦 the same. Here, we will multiply the first equation through by 3 to make the coefficients of 𝑥 the same in both equations to get 3𝑥3𝑦=24,3𝑥5𝑦=10.

We can now subtract the second equation from the first equation to eliminate the variable 𝑥. When doing this, we need to be particularly careful with our signs as we are subtracting negative numbers. We have 3𝑥3𝑥=0, (3𝑦)(5𝑦)=2𝑦, and 24(10)=34, giving us 2𝑦=34.

Dividing through by 2 gives us 𝑦=17.

In order to find the value of 𝑥, we need to substitute our value of 𝑦 into one of the original equations. Here, we will substitute into the first equation to get 𝑥17=8.

Adding 17 to each side gives us 𝑥=25.

The solution to this system of equations is, therefore, 𝑥=25 and 𝑦=17, which corresponds to the point (25,17).

Key Points

The steps to solving a system of two linear equations using elimination are as follows:

  1. Identify if the system contains a pair of coefficients of either unknown that have equal magnitude.
  2. If the system does not have a pair of coefficients of equal magnitude, multiply each equation through by a constant to make a pair of coefficients have equal magnitude.
  3. Once the system has a pair of coefficients with equal magnitude, add or subtract the equations to eliminate the variable.
  4. Solve the resulting equation to find one of the unknowns.
  5. Substitute this value back into one of the original equations to find the other unknown.

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