Solving Linear Equations With Substitution

Instructor: Gerald Lemay

Gerald has taught engineering, math and science and has a doctorate in electrical engineering.

Solving for the unknowns in a set of linear equations is a common task in math, science and engineering. In this lesson, we look at the substitution method as a systematic approach to finding the unknown values.

Solving Linear Equations with Substitution

Time for a new refrigerator! The selection is down to two units: same size but one costs $230 more. However, the more expensive unit is also more efficient. At some point in time, the money saved with the more efficient unit will offset the higher cost.

In this lesson, we will solve for two unknowns in two linear equations using the substitution method. This will help us decide which refrigerator to buy.

Choosing a refrigerator

The Substitution Method

Given two equations in two unknowns, the task is to determine the unknowns so both equations are true. In the substitution method, we choose one of the unknowns and make it the subject in one of the equations. We then substitute the right-hand side of this modified equation into the other equation and solve for the remaining unknown. Then, this value is substituted back into the modified equation to find the remaining unknown. Simplifying whenever possible is a good idea.

Let's do an example together.

Solve for x and y given 2x + 3y = 8 and 3x - 2y = -1.

  • Step 1: Choose an unknown and make it the subject in one of the equations (this is the modified equation)

Choosing x to be the subject in the first equation.

From 2x + 3y = 8,

subtract 3y from both sides: 2x + 3y - 3y = 8 - 3y

simplify: 2x = 8 - 3y

divide both sides by 2:

2x/2 = (8 - 3y)/2

simplify: x = 4 - 1.5y (our modified equation)

  • Step 2: Substitute the right-hand side into the other equation.

From 3x - 2y = -1, substitute 4 - 1.5y for x:

3(4 - 1.5y) - 2y = -1

simplify: 12 - 4.5y - 2y = -1

simplify: 12 - 6.5y = -1

  • Step 3: Get a value for the other unknown.

From 12 - 6.5y = -1,

subtract 12 from both sides: 12 - 6.5y - 12 = -1 -12

simplify: -6.5y = -13

divide both sides by -6.5: -6.5y/(-6.5) = -13/(-6.5)

simplify: y = 2

  • Step 4: Substitute this value into the modified equation and get a value for the first unknown.

From x = 4 - 1.5y, substitute y = 2:

x = 4 - 1.5(2)

simplify: x = 4 - 3 = 1

  • Step 5: (optional but a good idea) - Substitute the answers into the original equations as a check.

Substitute the answers into 2x + 3y = 8.

2(1) + 3(2) = 2 + 6 = 8 Check!

Substitute the answers into 3x - 2y = -1.

3(1) - 2(2) = 3 - 4 = -1 Check!

The answer using the substitution method is x = 1 and y = 2.

Now, let's return to selecting a refrigerator.

Deciding on a Refrigerator

In this refrigerator example, x is the number of months the refrigerator is used and y is the total cost. The total cost depends on the selling price of the refrigerator as well as the average monthly energy cost. The total cost y does not include maintenance. A new refrigerator could be maintenance free for 10 years.

Let's look at the available information for both refrigerators. The selling price for refrigerator 1 is $1600 and the average energy cost is $610 per year. For refrigerator 2, the selling price is $1830 and the average energy cost is $564 per year. Without getting into the details, this information leads to equations relating x and y for each refrigerator.

For refrigerator 1:

12y - 610x = 19200

For refrigerator 2:

3y - 141x = 5490

We now solve these two equations using the substitution method.

  • Step 1: Choose an unknown and make it the subject in one of the equations.

Choosing y to be the subject in the second equation;

From 3y - 141x = 5490,

add 141x to both sides: 3y - 141x + 141x = 5490 + 141x

simplify: 3y = 5490 + 141x

divide both sides by 3:

3y/3 = (5490 + 141x)/3

simplify: y = 5490/3 + 141x/3 = 1830 + 47x

Thus, we have y as the subject in the second equation: y = 1830 + 47x (the modified equation).

  • Step 2: Substitute the right-hand side into the other equation.

From 12y - 610x = 19200, substitute 1830 + 47x for y:

12(1830 + 47x) - 610x = 19200

simplify by distributing the multiply by 12:

12(1830) + 12(47x) - 610x = 19200

simplify: 21960 + 564x - 610x = 19200

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