Substitution Method

The Substitution Method is an algebraic method for finding the solutions of a system of equations. It consists of substituting an equivalent expression for a variable in one of the equations of the system. Consider, for example, the following system of linear equations. y-4=2x & (I) 9x+6=3y & (II) To solve the system by using the Substitution Method, there are four steps to follow.

Isolate One Variable in Any of the Equations

The first step is to isolate any variable in any of the equations. For simplicity, in this case, the y-variable will be isolated in Equation (I).

y-4=2x 9x+6=3y

AddEqn

(I): LHS+4=RHS+4

y=2x+4 9x+6=3y

Substitute the Expression

Substitute the new expression for the variable in the equation where the variable was not isolated. In this case, 2x+4 will be substituted for y in Equation (II).

y=2x+4 9x+6=3y

Substitute

(II): y= 2x+4

y=2x+4 9x+6=3( 2x+4)

Now Equation (II) only has one variable, which is x.

Solve the Equation With One Variable

Solve the equation that contains only one variable. In this case, Equation (II) will be solved for x.

y=2x+4 & (I) 9x+6=3(2x+4) & (II)

Distr

(II): Distribute 3

y=2x+4 9x+6=6x+12

SubEqn

(II): LHS-6=RHS-6

y=2x+4 9x=6x+6

SubEqn

(II): LHS-6x=RHS-6x

y=2x+4 3x=6

DivEqn

(II): .LHS /3.=.RHS /3.

y=2x+4 x=2

The value of the x-variable is 2.

Substitute the Value of the Variable Into the Other Equation

Now that the value of one of the variables is known, it can be substituted into the equation that has not been considered yet. Here, x= 2 will be substituted into Equation (I).

y=2x+4 & (I) x=2 & (II)

Substitute

(I): x= 2

y=2( 2)+4 x=2

▼

(I): Evaluate right-hand side

Multiply

(I): Multiply

y=4+4 x=2

AddTerms

(I): Add terms

y=8 x=2

The value of the y-variable in this system is 8. Therefore, the solution to the system of equations, which is the point of intersection of the lines, is (2,8) or x=2, y=8.

If at any step of the method a true statement is obtained, then the lines represented by the equations of the system are coincidental and the system will therefore have infinitely many solutions. Conversely, if at any step a false statement is obtained, then the lines are parallel and the system will have no solution.

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