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 Multiplication

Objective Learn how to use the Multiplication Property of Equality to modify systems of equations and then to solve them by using elimination by addition or subtraction.

In this lesson, you will learn a method that can be used to solve all systems of linear equations. The methods introduced so far only work in special cases, such as when the coefficient of one variable is 1, or when the coefficient of a variable in one equation is equal to or opposite of the coefficient in the other.


Elimination Using Multiplication

Recall the three methods for solving systems of equations that students have learned so far. Point out that each method has advantages and disadvantages.

(A) Solving by Graphing This method can be used for all systems, but produces only approximate solutions.

(B) Solving by Substitution This method can be used when the coefficient of one of the variables in one of the equations is  1.

(C) Elimination by Addition or Subtraction This method works when the coefficients of one variable are equal or opposite.

Now let's develop an additional method, which will allow us to solve all systems of linear equations. Look at the following system of equations.

3x + 5y = 11

6x + 4y = 16

This system of equations cannot easily be solved by substitution, since none of the coefficients is 1 or -1. Also, it cannot be solved using elimination by addition or subtraction. In this case, use the Multiplication Property of Equality to multiply the first equation by 2.

The reason for multiplying by 2 is that this makes the coefficient of x in the first equation equal to 6, which is the coefficient of x in the second equation. Then the x values can be eliminated by subtracting the second equation from the first. Another way to solve this system would be to multiply the first equation by -2 and then add the equations.

After multiplying by 2, the first equation becomes

6x + 10y = 22.

Now subtract the second equation from this modified first equation.

6x + 10y = 22  
( - )6x + 4y = 16 Subtract the equations.
0 + 6y = 6  
6y = 6  
y = 1 Divide each side by 6.

Now substitute 1 for y in, say, the first equation.

3x + 5(1) = 11 Substitute 1 for y.
3x + 5 = 11  
3x = 6 Subtract 5 from each side.
x = 2 Divide each side by 3.

The solution is (2, 1).

Key Idea

Any equation in a system of equations can be multiplied by a nonzero number to obtain an equivalent equation.

Use this fact to complete the following exercises.



Use elimination to solve each system of equations.

1. 3x + 6y = 15 2. 4x - 3y = 14 3. 1.5x + 2.5y = 4
  2x + 7y = 13   7x + 2y = 39   0.5x - 3.5y = -3
  (3, 1)   (5, 2)   (1, 1)




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