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TUTORIALS:

Absolute Values
Solving Two-Step Equations Algebraically
Multiplying Monomials
Factoring Trinomials
Solving Quadratic Equations
Power Functions and Transformations
Composition of Functions
Rational Inequalities
Equations of Lines
Graphing Logarithmic Functions
Elimination Using Multiplication
Multiplying Large Numbers
Multiplying by 11
Graphing Absolute Value Inequalities
Polynomials
The Discriminant
Reducing Numerical Fractions to Simplest Form
Addition of Algebraic Fractions
Graphing Inequalities in Two Variables
Adding and Subtracting Rational Expressions with Unlike Denominators
Multiplying Binomials
Graphing Linear Inequalities
Properties of Numbers and Definitions
Factoring Trinomials
Relatively Prime Numbers
Point
Inequalities
Rotating a Hyperbola
Writing Algebraic Expressions
Quadratic and Power Inequalities
Solving Quadratic Equations by Completing the Square
BEDMAS & Fractions
Solving Absolute Value Equations
Writing Linear Equations in Slope-Intercept Form
Adding and Subtracting Rational Expressions with Different Denominators
Reducing Rational Expressions
Solving Absolute Value Equations
Equations of a Line - Slope-intercept form
Adding and Subtracting Rational Expressions with Unlike Denominators
Solving Equations with a Fractional Exponent
Simple Trinomials as Products of Binomials
Equivalent Fractions
Multiplying Polynomials
Slope
Graphing Equations in Three Variables
Properties of Exponents
Graphing Linear Inequalities
Solving Cubic Equations by Factoring
Adding and Subtracting Fractions
Multiplying Whole Numbers
Straight Lines
Solving Absolute Value Equations
Solving Nonlinear Equations
Factoring Polynomials by Finding the Greatest Common Factor
Logarithms
Algebraic Expressions Containing Radicals 1
Addition Property of Equality
Three special types of lines
Quadratic Inequalities That Cannot Be Factored
Adding and Subtracting Fractions
Coordinate System
Solving Equations
Factoring Polynomials
Solving Quadratic Equations
Multiplying Radical Expressions
Solving Quadratic Equations Using the Square Root Property
The Slope of a Line
Square Roots
Adding Polynomials
Arithmetic with Positive and Negative Numbers
Solving Equations
Powers and Roots of Complex Numbers
Adding, Subtracting and Finding Least Common Denominators
What the Factored Form of a Quadratic can tell you about the graph
Plotting a Point
Solving Equations with Variables on Each Side
Finding the GCF of a Set of Monomials
Completing the Square
Solving Equations with Radicals and Exponents
Solving Systems of Equations By Substitution
Adding and Subtracting Rational Expressions
Percents
Laws of Exponents and Dividing Monomials
Factoring Special Quadratic Polynomials
Radicals
Solving Quadratic Equations by Completing the Square
Reducing Numerical Fractions to Simplest Form
Factoring Trinomials
Writing Decimals as Fractions
Using the Rules of Exponents
Evaluating the Quadratic Formula
Rationalizing the Denominator
Multiplication by 429
Writing Linear Equations in Point-Slope Form
Multiplying Radicals
Dividing Polynomials by Monomials
Factoring Trinomials
Introduction to Fractions
Square Roots
   
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Solving Systems of Equations By Substitution

After studying this lesson, you will be able to:

  • Solve systems of equations by substitution.

To Solve a System of Equations by Substitution:

1. Solve either equation for a variable. (Hint: Try to solve for a variable that does not have a coefficient.)

2. Substitute on equation into the other equation. This will eliminate one variable.

3. Solve for the remaining variable.

4. Substitute the solution into the other equation and solve for the other variable.

 

Example 1

Solve x + y = 6, x = y + 2

1 st : The second equation is already solved for a variable so we don't have to do anything in this step.

2 nd : Now we substitute x = y + 2 into the first equation. Since we know that x is the same as y + 2, we can substitute y + 2 for x in the first equation:

( y + 2 ) + y = 6

3 rd : Solve for y which is the remaining variable:

2y + 2 = 6

2y = 4

y = 2

4 th : Now we substitute the solution ( y = 2 ) into the second equation:

x = y + 2

x = (2) + 2

x = 4

The solution is (4, 2)

 

Example 2

Solve 2x + y = 13, 4x -3y = 11

1 st : We need to solve one equation for a variable. Let's solve the first equation for y since that is the variable without a coefficient:

2x + y = 13

y = 13 - 2x

2 nd : Now we substitute y = 13 - 2x into the second equation. Since we know that y is the same as 13 - 2x, we can substitute 13 - 2x for y in the second equation:

4x - 3y = 11

4x - 3 (13 - 2x ) = 11 (making the substitution)

3 rd : Solve for x which is the remaining variable:

4x - 3 (13 - 2x ) = 11

4x - 39 + 6x = 11

10x -39 = 11

10x = 50

x = 5

4 th : Now we substitute the solution ( x = 5 ) into the first equation:

2x + y = 13

2 (5) + y = 13

10 + y = 13

y = 3

The solution is (5, 3)

Special Cases: If you end up with a system where all variables cancel out, you have what we might call a "special case".

If the statement you're left with is true, the solution will be

If the statement you're left with is false, the solution will be Ø

 

Example 3

Solve x - 3y = -6, x - 3y = 6

1 st : We need to solve one equation for a variable. Let's solve the first equation for x since that is the variable without a coefficient:

x - 3y = -6

x = -6 + 3y

2 nd : Now we substitute x = -6 + 3y into the second equation.

x - 3y = 6

(-6 + 3y) -3y = 6 (making the substitution)

3 rd : Solve for y which is the remaining variable:

(-6 + 3y) -3y = 6

-6 = 6 All variables cancelled out so we have a special case.

Since the remaining statement is false, the solution is

 

Example 4

Solve x + 2y = 5, 3x - 15 = -6y

1 st : We need to solve one equation for a variable. Let's solve the first equation for x since that is the variable without a coefficient:

x + 2y = 5

x = 5 - 2y

2 nd : Now we substitute x = 5 - 2y into the second equation.

3x - 15 = -6y

3 (5 - 2y) - 15 = -6y (making the substitution)

3 rd : Solve for y which is the remaining variable:

3 (5 - 2y) - 15 = -6y

15 - 6y -15 = -6y

-6y = -6y

-6 = -6 All variables cancelled out so we have a special case.

Since the remaining statement is true, the solution is Ø

 

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Tuesday 19th of March