If you add the equations above, or add the opposite of one of the equations, you will get an equation that still has two variables. So let’s now use the multiplication property of equality first. You can multiply both sides of one of the equations by a number that will allow you to eliminate the same variable in the other equation.
We do this with multiplication. Notice that the first equation contains the term [latex]4y[/latex], and the second equation contains the term [latex]y[/latex]. If you multiply the second equation by [latex]−4[/latex], when you add both equations the y variables will add up to [latex]0[/latex].
The following example takes you through all the steps to find a solution to this system.
Example
Solve for [latex]x[/latex] and [latex]y[/latex].
Equation A: [latex]3x+4y=52[/latex]
Equation B: [latex]5x+y=30[/latex]
Show Solution
Look for terms that can be eliminated. The equations do not have any x or y terms with the same coefficients.
Caution! When you use multiplication to eliminate a variable, you must multiply EACH term in the equation by the number you choose. Forgetting to multiply every term is a common mistake.
Example
Solve the given system of equations by the elimination method.
[latex]\begin{array}{l}3x+5y=-11\hfill \\ x - 2y=11\hfill \end{array}[/latex]
Show Solution
Adding these equations as presented will not eliminate a variable. However, we see that the first equation has [latex]3x[/latex] in it and the second equation has [latex]x[/latex]. So if we multiply the second equation by [latex]-3,\text{}[/latex] the x-terms will add to zero.
Below is another video example of using the elimination method to solve a system of linear equations in which we multiply one of the equations be a constant.
It is worth demonstrating that there is more than one way to solve a system. Consider our first example. Instead of multiplying one equation in order to eliminate a variable when the equations were added, we could have multiplied both equations by different numbers.
Let’s remove the variable [latex]x[/latex] this time. Multiply Equation A by [latex]5[/latex] and Equation B by [latex]−3[/latex].
In order to use the elimination method, you have to create variables that have the same coefficient—then you can eliminate them. Multiply the top equation by [latex]5[/latex].
These equations were multiplied by [latex]5[/latex] and [latex]−3[/latex] respectively, because that gave you terms that would add up to [latex]0[/latex]. Be sure to multiply all of the terms of the equation.
In the next example, we will see that sometimes we have to multiply both numbers by different numbers in order for one variable to be eliminated.
Example
Solve the given system of equations in two variables by elimination.
One equation has [latex]2x[/latex] and the other has [latex]5x[/latex]. The least common multiple is [latex]10x[/latex], so we will have to multiply both equations by a constant in order to eliminate one variable. Let’s eliminate [latex]x[/latex] by multiplying the first equation by [latex]-5[/latex] and the second equation by [latex]2[/latex].
Below is a summary of the general steps for using the elimination method to solve a system of equations.
How To: Given a system of equations, solve using the elimination method
Write both equations with x and y-variables on the left side of the equal sign and constants on the right.
Write one equation above the other, lining up corresponding variables. If one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, add the equations together, eliminating one variable. If not, use multiplication by a nonzero number so that one of the variables in the top equation has the opposite coefficient of the same variable in the bottom equation, then add the equations to eliminate the variable.
Solve the resulting equation for the remaining variable.
Substitute that value into one of the original equations and solve for the second variable.
Check the solution by substituting the values into the other equation.
Try It
In the next example, we will show how to solve a system with fractions. As with single linear equations, the easiest way to solve is to clear the fractions first with the least common denominator.
Example
Solve the given system of equations in two variables by elimination.
In the following video, you will find one more example of using the elimination method to solve a system; this one has coefficients that are fractions.
It is possible to use the elimination method with multiplication and get a result that indicates no solutions or infinitely many solutions, just as with the other methods we have learned for finding solutions to systems. Recall that a dependent system of equations in two variables is a system in which the two equations represent the same line. Dependent systems have an infinite number of solutions because all of the points on one line are also on the other line. After using substitution or elimination, the resulting equation will be an identity such as [latex]0=0[/latex]. The last example includes two equations that represent the same line and are therefore dependent.
Example
Find a solution to the system of equations using the elimination method.
With the elimination method, we want to eliminate one of the variables by adding the equations. In this case, focus on eliminating [latex]x[/latex]. If we multiply both sides of the first equation by [latex]-3[/latex], then we will be able to eliminate the [latex]x[/latex] -variable.
We can see that there will be an infinite number of solutions that satisfy both equations.
If we rewrote both equations in slope-intercept form, we might know what the solution would look like before adding. Look at what happens when we convert the system to slope-intercept form.
See the graph below. Notice the results are the same. The general solution to the system is [latex]\left(x, -\dfrac{1}{3}x+\dfrac{2}{3}\right)[/latex].
In the following video, the elimination method is used to solve a system of equations. Notice that one of the equations needs to be multiplied by a negative one first. Additionally, this system has an infinite number of solutions.
In our last video example, we present a system that is inconsistent; it has no solutions which means the lines the equations represent are parallel to each other.
Try It
Summary
Multiplication can be used to set up matching terms in equations before they are combined to aid in finding a solution to a system. When using the multiplication method, it is important to multiply all the terms on both sides of the equation—not just the one term you are trying to eliminate.
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Ex: System of Equations Using Elimination (Infinite Solutions). Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/NRxh9Q16Ulk. License: CC BY: Attribution
Unit 14: Systems of Equations and Inequalities, from Developmental Math: An Open Program. Provided by: Monterey Institute of Technology and Education. Located at: . License: CC BY: Attribution
Ex 2: Solve a System of Equations Using the Elimination Method. Authored by: James Sousa (Mathispower4u.com) for Lumen Learning. Located at: https://youtu.be/_liDhKops2w. License: CC BY: Attribution