Reading: The Substitution Method

Learning Objective(s)

·Solve a system of equations using the substitution method.

·Recognize systems of equations that have no solution or an infinite number of solutions.

·Solve application problems using the substitution method.

A graph can be used to show the solution for a system of two linear equations. However, accurately determining the solution from a graph is not always easy or accurate. For example, where do you think the two lines shown below intersect?

It looks like they might intersect at (1.8, -0.7)--though this is only an estimate. In cases like this, you can use algebraic methods to find exact answers. One method to look at is called the substitution method. You solve one equation for one variable and then substitute this expression into the other equation.

In the substitution method you solve for one variable, and then substitute that expression into the other equation. The important thing here is that you are always substituting values that are equivalent.

For example:

Sean is 5 years older than four times his daughter's age. His daughter is 7. How old is Sean?

You might do this problem in your head. Sean's daughter is 7, so "four times his daughter's age” is 28, and 5 years added to that is 33. Sean is 33.

If you solved the problem like that, you used a simple substitution--you substituted in the value "7” for "his daughter's age.” You learned in the second part of the problem that "his daughter is 7.” So substituting in a value of "7” for "his daughter's age” in the first part of the problem was okay, because you knew these two quantities were equal.

Let's look at a simple system of equations that can be solved using substitution.

You can substitute a value for a variable even if it is an expression. Here's an example.

Remember, a solution to a system of equations must be a solution to each of the equations within the system. The ordered pair (4, −1) does work for both equations, so you know that it is a solution to the system as well.

Let's look at another example whose substitution involves the distributive property.

In the examples above, one of the equations was already given to us in terms of the variable x or y. This allowed us to quickly substitute that value into the other equation and solve for one of the unknowns.

Sometimes you may have to rewrite one of the equations in terms of one of the variables first before you can substitute. Look at the example below.

There are some cases where using the substitution method will yield results that, at first, do not make sense. Let's take a look at some of these and figure out what is going on.

You get the false statement −8 = 4. What does this mean? The graph of this system sheds some light on what is happening.

The lines are parallel, they never intersect and there is no solution to this system of linear equations. Note that the result −8 = 4 is not a solution. It is simply a false statement and it indicates that there is no solution.

Now take this problem, which is interesting as well.

Solve for x and y.

y = −0.5x

9y = −4.5x

Substituting −0.5x for y in the second equation, you find the following:

This time you get a true statement: −4.5x = −4.5x. But what does this type of answer mean? Again, graphing can help you make sense of this system.

This system consists of two equations that both represent the same line; the two lines are collinear. Every point along the line will be a solution to the system, and that's why the substitution method yields a true statement. In this case, there are an infinite number of solutions.

Systems of equations are a very useful tool for modeling real-life situations and answering questions about them. If you can translate the application into two linear equations with two variables, then you have a system of equations that you can solve to find the solution. You can use any method to solve the system of equations. Use the substitution method in this topic.

In order to sell more of its produce, a local farm sells bags of apples in two sizes: medium and large. A medium bag contains 4 Macintosh and 1 Granny Smith apples and costs $2.80. A large bag contains 8 Macintosh and 4 Granny Smith apples and costs $7.20. The price of one Granny Smith apple is the same in the medium bag as it is in the large bag. The price of one Macintosh apple is the same in the medium bag as it is in the large bag. What is the price of each kind of apple?

Let's start by creating a system of equations that represents what is happening in the problem. There are two types of apples and two sizes of bags. You can let m represent the cost of a Macintosh apple and g represent the price of a Granny Smith apple. Let's make a table and see what is known.

Now that you have two equations in the same variables you can solve the system. You will use substitution. The steps are shown in the example below:

Using the substitution method can be an effective approach to solving geometric problems.

The substitution method is one way of solving systems of equations. To use the substitution method, use one equation to find an expression for one of the variables in terms of the other variable. Then substitute that expression in place of that variable in the second equation. You can then solve this equation as it will now have only one variable. Solving using the substitution method will yield one of three results: a single value for each variable within the system (indicating one solution), an untrue statement (indicating no solutions), or a true statement (indicating an infinite number of solutions).

This reading is taken from the Developmental Math Open Program created by The NROC Project. It is available under a Creative Commons license.