FreeAlgebra Tutorials!   Home 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 Powers 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 http: 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 Fractions 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 Adding Quadratic Functions Conjugates Factoring 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 Percents Arithmetics 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 Polynomials Laws of Exponents Multiplying Polynomials Vertical Line Test http: Solving Inequalities with Fractions and Parentheses http: Multiplying Polynomials Fractions Solving Quadratic and Polynomial Equations Extraneous Solutions Fractions

Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# 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.

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)