Elimination Using Multiplication
Prime Factors
Equations Involving Rational Exponents
Working with Percentages and Proportions
Rational Expressions
Interval Notation and Graphs
Simplifying Complex Fractions
Dividing Whole Numbers with Long Division
Solving Compound Linear Inequalities
Raising a Quotient to a Power
Solving Rational Equations
Solving Inequalities
Adding with Negative Numbers
Quadratic Inequalities
Dividing Monomials
Using the Discriminant in Factoring
Solving Equations by Factoring
Subtracting Polynomials
Cube Root
The Quadratic Formula
Multiply by the Reciprocal
Relating Equations and Graphs for Quadratic Functions
Multiplying a Polynomial by a Monomial
Calculating Percentages
Solving Systems of Equations using Substitution
Comparing Fractions
Solving Equations Containing Rational Expressions
Factoring Polynomials
Negative Rational Exponents
Roots and Radicals
Intercepts Given Ordered Pairs and Lines
Factoring Polynomials
Solving Linear Inequalities
Mixed Expressions and Complex Fractions
Solving Equations by Multiplying or Dividing
The Addition Method
Finding the Equation of an Inverse Function
Solving Compound Linear Inequalities
Multiplying and Dividing With Square Roots
Exponents and Their Properties
Equations as Functions
Factoring Trinomials
Solving Quadratic Equations by Completing the Square
Dividing by Decimals
Lines and Equations
Simplifying Complex Fractions
Graphing Solution Sets for Inequalities
Standard Form for the Equation of a Line
Checking Division with Multiplication
Elimination Using Addition and Subtraction
Complex Fractions
Multiplication Property of Equality
Solving Proportions Using Cross Multiplication
Product and Quotient of Functions
Quadratic Functions
Solving Compound Inequalities
Operating with Complex Numbers
Equivalent Fractions
Changing Improper Fractions to Mixed Numbers
Multiplying by a Monomial
Solving Linear Equations and Inequalities Graphically
Dividing Polynomials by Monomials
Multiplying Cube Roots
Operations with Monomials
Properties of Exponents
Mixed Numbers and Improper Fractions
Equations Quadratic in Form
Simplifying Square Roots That Contain Whole Numbers
Dividing a Polynomial by a Monomial
Writing Numbers in Scientific Notation
Solutions to Linear Equations in Two Variables
Solving Linear Inequalities
Multiplying Two Mixed Numbers with the Same Fraction
Special Fractions
Solving a Quadratic Inequality
Parent and Family Graphs
Solving Equations with a Fractional Exponent
Evaluating Trigonometric Functions
Solving Equations Involving Rational Expressions
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Elimination Using Addition and Subtraction

Objective Introduce the elimination method of solving systems of simultaneous equations.

The main idea in this lesson is that systems of equations may be simplified and solved using the Addition and Multiplication Properties of Equality to add and to subtract equations. The goal is to obtain an equivalent equation in which only one variable is on the left-hand side and only a number is on the right-hand side.


The Elimination Method for Solving Systems of Equations

Let's start with some examples.

Example 1

Solve the system of linear equations.

x - y = 3

x + y = 7


For (x , y ) to be a solution to the system means that both equations hold true for the values (x , y ). That is, both equations are satisfied. Add the equations to get another valid equation, using the Addition Property of Equality.

x - y = 3  
(+) x + y = 7 Add the equations.
2x + 0 = 10  
2x = 10  

This is a linear equation in one variable, namely x. Solve to get x = 5. Then substitute x = 5 into one of the original equations and solve for y.

x - y = 3  
5 - y = 3 Replace x with 5.
-y = -2  
y = 2  

The only solution to the system of equations is (5, 2).


Example 2

Solve the system of equations.

2x + 5y = 7

2x - 2 y = 0


Use the Subtraction Property of Equality to subtract one of the equations from the other to get another valid equation.

2x + 5y = 7  
(-) 2x - 2 y = 0 Subtract the equations.
0 + 7y = 7  

Solve 7y = 7 to get y = 1. Then substitute the value of y into one of the equations and solve for x.

2x - 2(1) = 0 Substitute 1 for y.
2x - 2 = 0  
2x = 2  
x = 1 Divide each side by 2.

The solution is (1, 1).

When the resulting equation involves only one variable, we say that we have eliminated the other variable. In Example 1, y was eliminated, and in Example 2, x was eliminated. Note to the Teacher It is important for students to practice using this method by working the exercises below. Then, introduce Examples 3 and 4, which are word problems.


Example 3

The sum of two numbers is 48, and their difference is 16. What are the numbers?


Let x and y represent the two numbers and write the following system of equations.

x + y = 48

x - y = 16

Adding the equations gives 2x = 64 or x = 32. To find the value of y, substitute 32 for x in either equation to get y = 16. The two numbers are 32 and 16.



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