We can use the Properties of Equality in solving systems of linear equations algebraically to eliminate a variable. There are several steps to follow in order to eliminate a variable in a system of equations. We will present these steps and the possible Properties of Equality that can be used in these steps in a table.
What is done?
What properties can be used?
Step I
Gather all like terms on the same sides of the equations.
Solve the equation for the other variable by substituting the obtained variable into one of the equations in the system.
Substitution Property of Equality and other Properties of Equality used to solve an equation
Let's give an example.
Example
Consider the following system of linear equations.
4x+2y=63 & (I) 3x=3.5+y & (II)
Step I
In Equation (I), the variable terms are on the same side of the equation. However, in Equation (II) the variable terms are on both sides of the equations. We can apply the Subtraction Property of Equality to gather the variables on the left-hand side of the equation.
3x - y=3.5+y - y
⇕
3x-y=3.5
Step II
Using the Multiplication Property of Equality, we will multiply both sides of Equation (II) by 2 to have opposite coefficients for the y-variable.
2 (3x-y)= 2(3.5)
⇕
6x-2y= 7
Step III
We will add the obtained equation to Equation (I) by using the Addition Property of Equation.
We will now substitute x = 7 into one of the equations by using the Substitution Property of Equality. Let's use Equation (I). We will then apply the Subtraction and Division Properties of Equality to solve the equation for y.
