This document provides an overview of solving systems of linear equations algebraically. There are two main methods: substitution and elimination. Substitution involves isolating one variable and substituting it into the other equation. Elimination involves adding or subtracting equations to eliminate one variable. Examples are provided to demonstrate both methods. The objectives are for students to be able to solve linear systems algebraically and relate it to representing relationships between quantities with graphs and equations.
2. Essential Understanding and
Objectives
• Essential Understanding: you can solve a system of equations by
writing equivalent systems until the value of one variable is clear.
Then substitute to find the values of the other variable
• Objectives:
• Students will be able to solve linear systems algebraically
3. Iowa Core Curriculum
• Algebra
• A. CED.2 Create equations in two or more variables to represent
relationships between quantities; graph equations on coordinate
axes with labels and scales.
• A.CED.3 . Represent constraints by equations or inequalities, and by
systems of equations and/or inequalities, and interpret solutions as
viable or nonviable options in a modeling context. For example,
represent inequalities describing nutritional and cost constraints on
combinations of different foods.
• A.REI.5 Prove that, given a system of two equations in two variables,
replacing one equation by the sum of that equation and a multiple
of the other produces a system with the same solutions.
• A.REI.6 Solve systems of linear equations exactly and approximately
(e.g., with graphs), focusing on pairs of linear equations in two
variables.
4. Substitution Method
• Use this method to solve a system of equations when it is easy
to isolate one of the variables.
• After isolating one of the variables, substitute for that variable
in the other equation.
• Then solve for the other variable
5. Example
• What is the solution to the system of equations?
ì3x + 4y = 12
í
î2x + y = 10
• Step 1: solve the equation for one of the variables
• Step 2: Substitute the expression for y in the other equation.
Then solve for x
• Step 3: Substitute the value for x into one of the original
equations. Solve for y.
6. Example
• What is the solution of the system of equations?
ì x + 3y = 5
í
î-2x - 4y = -5
7. Example
• An online music company offers 15 downloads for $19.75 and 40
downloads for $43.50. Each price includes the same one-time
registration fee. What is the cost of each download and the
registration fee?
• Step 1: Relate
• 15 (cost of one download) + one-time fee = $19.75
• 40 (cost of one download) + one-time fee = $43.50
• Step 2: Define
• Let c = cost of one download
• Let f = the one time fee
• Step 3: Write
• 15c+ f = 19.75
• 40c + f= 43.50
• Step 4: Solve using substitution
• Solution: $.95 per download and $5.50 for the one time fee
8. Elimination Method
• You can use the Addition Property of Equality (If a = b then a
+ c = b + c )to solve a system of equations. By adding a pair of
additive inverses or subtract identical terms you can eliminate
a variable.
9. Example
• What is the solution of the system of equations?
ì 4x + 2y = 9
í
î-4x + 3y = 16
• Step 1: Do you need to multiply to get equivalent terms?
• Step 2: Add the equations together
• Step 3: Solve for the remaining variable
• Step 4: Chose one of the original equations and substitute the
variable you already solved for.
• Step 5: Solve for the other variable
10. Example
• What is the solution of the system of equations?
ì-2x + 8y = -8
í
î5x - 8y = 20
11. Equivalent Systems
• When you multiply each side of one or both equations in the
same system by the same nonzero number, the new system
and the original system has the same solutions. This is known
as Equivalent Systems
• Example 4
• What is the solution of the system of equations?
ì2x + 7y = 4
í
î3x + 5y = -5
12. What are the solutions of the
following systems? Explain.
ì5x + 2y = -4 ì-x + y = -2
í í
î3x + 7y = 15 î2x - 2y = 0
ì 4x + y = 6
í
î12x + 3y = 18
