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Mixed-Number Notation
Linear Equations and Inequalities in One Variable
The Quadratic Formula
Fractions and Decimals
Graphing Logarithmic Functions
Multiplication by 111
Fractions
Solving Systems of Equations - Two Lines
Solving Nonlinear Equations by Factoring
Solving Linear Systems of Equations by Elimination
Rationalizing the Denominator
Simplifying Complex Fractions
Factoring Trinomials
Linear Relations and Functions
Polynomials
Axis of Symmetry and Vertices
Equations Quadratic in Form
The Appearance of a Polynomial Equation
Subtracting Reverses
Non-Linear Equations
Exponents and Order of Operations
Factoring Trinomials by Grouping
Factoring Trinomials of the Type ax 2 + bx + c
The Distance Formula
Invariants Under Rotation
Multiplying and Dividing Monomials
Solving a System of Three Linear Equations by Elimination
Multiplication by 25
Powers of i
Solving Quadratic and Polynomial Equations
Slope-intercept Form for the Equation of a Line
Equations of Lines
Square Roots
Integral Exponents
Product Rule for Radicals
Solving Compound Linear Inequalities
Axis of Symmetry and Vertices
Multiplying Rational Expressions
Reducing Rational Expressions
Properties of Negative Exponents
Fractions
Numbers, Factors, and Reducing Fractions to Lowest Terms
Solving Quadratic Equations
Factoring Completely General Quadratic Trinomials
Solving a Formula for a Given Variable
Factoring Polynomials
Decimal Numbers and Fractions
Multiplication Properties of Exponents
Multiplying Fractions
Multiplication by 50


 
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Example

Use elimination to find the solution of this system.

x - 3y = -17 First equation

-x + 8y = 52 Second equation

Solution

Add the two equations.

 

 

x

- x

-

+

3y

8y

=

=

-

 

17

52

0x + 5y =   35

Simplify. The x-terms have been eliminated.

To solve for y, divide both sides by 5.

To find the value of x, substitute 7 for y in either of the original equations. Then solve for x.

5y

y

= 35

= 7

We will use the first equation.

Substitute 7 for y.

Multiply.

Add 21 to both sides.

The solution of the system is (4, 7).

To check the solution, substitute 4 for x and 7 for y into each original equation. Then simplify.

In each case, the result will be a true statement.

The details of the check are left to you.

x - 3y

x - 3(7)

x - 21

x

= -17

= -17

= -17

= 4

 

In the two original equations in the previous example, the coefficients of x were opposites. Thus, when the equations were added, the x-terms were eliminated.
1x

- 1x

-

+

3y

8y

=

=

-

 

17

52

    5y =   35

When the coefficients of neither variable are opposites, we choose a variable. Then we multiply both sides of one (or both) equations by an appropriate number (or numbers) to make the coefficients of that variable opposites.

Note:

The Multiplication Principle of Equality enables us to multiply both sides of an equation by the same nonzero number without changing the solutions of the equation.