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Finding the Greatest Common Factor
Factoring Trinomials
Absolute Value Function
A Summary of Factoring Polynomials
Solving Equations with One Radical Term
Adding Fractions
Subtracting Fractions
The FOIL Method
Graphing Compound Inequalities
Solving Absolute Value Inequalities
Adding and Subtracting Polynomials
Using Slope
Solving Quadratic Equations
Multiplication Properties of Exponents
Completing the Square
Solving Systems of Equations by using the Substitution Method
Combining Like Radical Terms
Elimination Using Multiplication
Solving Equations
Pythagoras' Theorem 1
Finding the Least Common Multiples
Multiplying and Dividing in Scientific Notation
Adding and Subtracting Fractions
Solving Quadratic Equations
Adding and Subtracting Fractions
Multiplication by 111
Adding Fractions
Multiplying and Dividing Rational Numbers
Multiplication by 50
Solving Linear Inequalities in One Variable
Simplifying Cube Roots That Contain Integers
Graphing Compound Inequalities
Simple Trinomials as Products of Binomials
Writing Linear Equations in Slope-Intercept Form
Solving Linear Equations
Lines and Equations
The Intercepts of a Parabola
Absolute Value Function
Solving Equations
Solving Compound Linear Inequalities
Complex Numbers
Factoring the Difference of Two Squares
Multiplying and Dividing Rational Expressions
Adding and Subtracting Radicals
Multiplying and Dividing Signed Numbers
Solving Systems of Equations
Factoring Out the Opposite of the GCF
Multiplying Special Polynomials
Properties of Exponents
Scientific Notation
Multiplying Rational Expressions
Adding and Subtracting Rational Expressions With Unlike Denominators
Multiplication by 25
Decimals to Fractions
Solving Quadratic Equations by Completing the Square
Quotient Rule for Exponents
Simplifying Square Roots
Multiplying and Dividing Rational Expressions
Independent, Inconsistent, and Dependent Systems of Equations
Graphing Lines in the Coordinate Plane
Graphing Functions
Powers of Ten
Zero Power Property of Exponents
The Vertex of a Parabola
Rationalizing the Denominator
Test for Factorability for Quadratic Trinomials
Trinomial Squares
Solving Two-Step Equations
Solving Linear Equations Containing Fractions
Multiplying by 125
Exponent Properties
Multiplying Fractions
Adding and Subtracting Rational Expressions With the Same Denominator
Quadratic Expressions - Completing Squares
Adding and Subtracting Mixed Numbers with Different Denominators
Solving a Formula for a Given Variable
Factoring Trinomials
Multiplying and Dividing Fractions
Multiplying and Dividing Complex Numbers in Polar Form
Power Equations and their Graphs
Solving Linear Systems of Equations by Substitution
Solving Polynomial Equations by Factoring
Laws of Exponents
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Systems of Linear Equations
Properties of Rational Exponents
Power of a Product and Power of a Quotient
Factoring Differences of Perfect Squares
Dividing Fractions
Factoring a Polynomial by Finding the GCF
Graphing Linear Equations
Steps in Factoring
Multiplication Property of Exponents
Solving Systems of Linear Equations in Three Variables
Solving Exponential Equations
Finding the GCF of a Set of Monomials
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Solving Systems of Equations by using the Substitution Method

Objective Learn to solve systems of equations by using the substitution method, and reinforce the geometric concept of what solving the system means.

Graph the systems after you have found the solutions so that the geometric idea that the solution corresponds to the intersection of lines is reinforced.


Solving Systems of Equations Algebraically

You should have solved systems of equations graphically by graphing the lines corresponding to the two equations and finding the coordinates of the point of intersection by “inspection”. Often this method is not good enough because it is sometimes difficult to get an exact answer. Instead, there is an algebraic method for solving systems of equations. Let's begin with an example.


Example 1

Find the solution to the system of linear equations.

3x + y = 3

2x + 5y = 6


You can graph the equations to get a rough idea of what the solution set is. The graph of this system of equations is shown below.

Use the graph to estimate the coordinates of the intersection point to be about (0.8, 0.9). So, the solution is approximately x = 0.8, y = 0.9. To find the exact solution, solve the system of equations algebraically.

Key Idea

Use the Addition and Multiplication Properties of Equality to solve systems of equations in two variables, just as equations with one variable are solved.

3x + y = 3  
y = 3 - 3x Subtract 3x from each side.

This equation is solved for y in terms of x . Since the value of y must be the same in both equations, substitute 3 - 3x for y in the second equation.

2x + 5y = 6  
2x + 5(3 - 3x ) = 6 Substitute 3 - 3x for y.
2x + 15 - 15x = 6 Distributive Property
-13x + 15 = 6  

This is an equation involving only one variable, namely x. Now solve the equation for x.

-13x + 15 = 6  
-13x = -9 Subtract 15 from each side.
x Divide each side by -13.

The exact value for the x-coordinate of the solution is . To find the exact value for the y-coordinate, substitute for x in either of the two equations.

Now use the Subtraction Property of Equality to solve for y.

So, the exact solution is given by

This method always works when the system has exactly one solution.

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