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Saturday 15th of June
Rotating a Parabola
Multiplying Fractions
Finding Factors
Miscellaneous Equations
Mixed Numbers and Improper Fractions
Systems of Equations in Two Variables
Literal Numbers
Adding and Subtracting Polynomials
Subtracting Integers
Simplifying Complex Fractions
Decimals and Fractions
Multiplying Integers
Logarithmic Functions
Multiplying Monomials
The Square of a Binomial
Factoring Trinomials
The Pythagorean Theorem
Solving Radical Equations in One Variable
Multiplying Binomials Using the FOIL Method
Imaginary Numbers
Solving Quadratic Equations Using the Quadratic Formula
Solving Quadratic Equations
Order of Operations
Dividing Complex Numbers
The Appearance of a Polynomial Equation
Standard Form of a Line
Positive Integral Divisors
Dividing Fractions
Solving Linear Systems of Equations by Elimination
Multiplying and Dividing Square Roots
Functions and Graphs
Dividing Polynomials
Solving Rational Equations
Use of Parentheses or Brackets (The Distributive Law)
Multiplying and Dividing by Monomials
Solving Quadratic Equations by Graphing
Multiplying Decimals
Use of Parentheses or Brackets (The Distributive Law)
Simplifying Complex Fractions 1
Adding Fractions
Simplifying Complex Fractions
Solutions to Linear Equations in Two Variables
Quadratic Expressions Completing Squares
Dividing Radical Expressions
Rise and Run
Graphing Exponential Functions
Multiplying by a Monomial
The Cartesian Coordinate System
Writing the Terms of a Polynomial in Descending Order
Quadratic Expressions
Solving Inequalities
Solving Rational Inequalities with a Sign Graph
Solving Linear Equations
Solving an Equation with Two Radical Terms
Simplifying Rational Expressions
Intercepts of a Line
Completing the Square
Order of Operations
Factoring Trinomials
Solving Linear Equations
Solving Multi-Step Inequalities
Solving Quadratic Equations Graphically and Algebraically
Collecting Like Terms
Solving Equations with Radicals and Exponents
Percent of Change
Powers of ten (Scientific Notation)
Comparing Integers on a Number Line
Solving Systems of Equations Using Substitution
Factoring Out the Greatest Common Factor
Families of Functions
Monomial Factors
Multiplying and Dividing Complex Numbers
Properties of Exponents
Multiplying Square Roots
Adding or Subtracting Rational Expressions with Different Denominators
Expressions with Variables as Exponents
The Quadratic Formula
Writing a Quadratic with Given Solutions
Simplifying Square Roots
Adding and Subtracting Square Roots
Adding and Subtracting Rational Expressions
Combining Like Radical Terms
Solving Systems of Equations Using Substitution
Dividing Polynomials
Graphing Functions
Product of a Sum and a Difference
Solving First Degree Inequalities
Solving Equations with Radicals and Exponents
Roots and Powers
Multiplying Numbers
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Solving Systems of Equations Using Substitution

Two algebraic methods for solving systems of equations are addition and substitution.

A “system of equations” is defined by the set of all of the lines that intersect at the point (h, k) and the solution which can be determined by the equations of two different lines. The simplest of these are { x = h, y = k } which is called the “solution set” of the system of equations. The solution is also defined by the point of intersection: { (h, k) }.

In any system of equations (family of lines) the following are true statements.

1. A linear equation may be multiplied by a real number and form another linear equation through the same point. (a coincident line).

2. Two intersecting linear equations may be added to form the equation of another line through the same point on a graph.

3. One equation can be “solved for one of the variables” and that expression can then be substituted in another equation in place of that variable.

NOTE: [Generally, use only for variables with coefficient of ±1.]


Substitution Method

Unless one of the equations is already solved for one of the variables in terms of the other, the Substitution Method is not generally as desirable as the Addition Method.


Example 1:

Since the second equation is already solved for y we can use the expression on the right side as a replacement for y in the first equation, which gives us an equation in x alone.


x + (2x - 1) = 11 Replacement for y
3x - 1 = 11 Distributive Property
3x = 12 Add Opps
x = 4 Multiply Recip

Replace x = 4 in the second equation y = 2( 4 ) − 1 or y = 7.

Check: Substitute both in equation one: ( 4 ) + ( 7 ) = 11 The solution point: { ( 4, 7) }


Example 2:

Since the first equation is already solved for y we can use the expression on the right side as a replacement for y in the second equation, which gives us an equation in x alone.

Substitute the expression for y in the second equation: 3x − 2(-x + 8) = − 1

Solve this equation for x:

3x + 2x − 16 = − 1 Distributive property
5x = 15 Add Opps
x = 3 Multiply Recip

Replace x = 3 in equation (1): y = − ( 3 ) + 8 finds y = 5 or solution point: { ( 3, 5) }

Always check in the other equation: 3(3) − 2(5) = − 1

Never consider using the substitution method if all of the coefficients in the system are different than ±1. If none are equal to ±1 you may have fractions to contend with and that’s just not necessary. Some simple systems do have fractions as solutions.

In fact, if you encounter a system of equations where the coefficients are fractions, it is usually better to multiply first by the LCM then proceed to solve the system.

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