Solving Systems of Equations

By Substitution

Objective

The student will solve systems of equations by substitution.

Essential Questions

• Why will solving for x first rather than for y give the same solution?

• After solving for one variable, how will you choose an equation to solve for the second value?

Methods Used to Solve Systems of Equations• Graphing• Substitution• Elimination (Linear Combination)

A Word About Substitution

• Substitution is a good method to use if one variable in one of the equations is already isolated or has a coefficient of one.

• Substitution can be used for systems of two or three equations, but many prefer other methods for three equation systems.

Let’s Work Some Problems Using

Substitution.

Substitution

The goal in substitution is to combine the twoequations so that there is just one equation with one variable.

Substitution

Solve the system using substitution.

y = 4x x + 3y = –39

x + 3(4x) = – 39

x + 12x = –39

13x = –39

x = – 3 Continued on next slide.

Since y is already isolated in the first equation,substitute the value of y for y in the second equation.

The result is one equation with one variable.

Substitution

After solving for x, solve for y by substitutingthe value for x in any equation that contains 2 variables.

y = 4x y = 4(–3) y = –12

Write the solution as an ordered pair. (–3, –12)

There’s more on the next slide.

Substitution

Check the solution in BOTH equations.

y = 4x x + 3y = –39

–12 = 4(–3) –12 = –12

–3 + 3(– 12) = –39 –3 – 36 = –39 –39 = –39

P

P

The solution is (– 3, –12).

Substitution

Solve the system using substitution.

x – 3y = –5 2x + 7y = 16

x = 3y – 5 2x + 7y = 16

2(3y – 5) + 7y = 16

If a variable is not already isolated, solve for one variable in one of the equations. Choose to solve for a variable with a coefficient of one,if possible.

Substitution

2(3y – 5) + 7y = 16

6y – 10 + 7y = 16

13y – 10 = 16

13y = 26

y = 2

x = 3y – 5 2x + 7y = 16

x = 3(2) – 5 x = 6 – 5 x = 1

The solution is (1, 2).

* Be sure to check!

Substitution

A zookeeper needs to mix feed for the prairie dogs So that the feed has the right amount of protein. Feed A has 12% protein. Feed B has 5% protein. How many pounds of each does he need to mix toget 100 lb of feed that is 8% protein?

Substitution