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
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
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
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
Quadratic Functions
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
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
Laws of Exponents
Multiplying Polynomials
Vertical Line Test
Solving Inequalities with Fractions and Parentheses
Multiplying Polynomials
Solving Quadratic and Polynomial Equations
Extraneous Solutions
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Solutions to Linear Equations in Two Variables

1. Given 2x − 3y = 6 build the table for using an arithmetic sequence as replacement values.

Logically, we know that it is easy to plot ordered pairs if both coordinates are integers.

In this equation we note that all of the numbers except the y-coefficient are even integers.

From arithmetic we know that:

Multiplying by any even integer will yield an even integer as the product


Adding two even integers will result in an even integer as the sum.

Even × Odd (or Even) = Even


Even ± Even = Even

Now if we choose an even integer as the replacement for the y- value then the resulting x-value will be an integer. Therefore, we put the "sequence of even integers" in the table for y then complete the computation below for each y-value and put each corresponding x-value in the table. Repeat the process for the other y-values to find the resulting x-values. Note any sequences and write the common differences.

2x − 3y = 6

Let y = - 4: 2x − 3(- 4) = 6,

repeat for the other numbers

y = - 2: 2x − 3(- 2) = 6, 

y = 0: 2x − 3( 0 ) = 6, 

y = 2: 2x − 3( 2 ) = 6, 

y = 4: 2x − 3( 4 ) = 6,


x = - 3


x = 0

x = 3

x = 6

x = 9

Check to see that both columns of values are arithmetic sequences.

For the point (- 3, - 4) replace x = - 3 and y = - 4 in the given equation 2x − 3y = 6


For the point (9, 4) replace x = 9 and y = 4 in the given equation 2x − 3y = 6


You should always form the practice of checking all of your work whenever you can.

2. Graph 2x + 3y = 7. Build a table for 2x + 3y = 7.

Again let us turn to LOGIC: We know that for integers:

Odd × Odd = Odd Odd ± Odd = Even Even × Odd (or Even) = Even Even ± Odd = Odd

In order to keep both coordinates as integers choose the replacement values for the y to be odd integers. We choose the sequence of odd integers for replacement for y- values and the resulting x-sequence will be integers. Complete the computation below and put the number in the table. Repeat the process for the other y-values to find the resulting x-values.

2x + 3y = 7      Let y = 1

 repeat for the other numbers

Put these values in the table, then write the “common differences” for both x and y.

Use the value for dx to complete the table values for x.


For complete table see below.

Note adding ± 3 to given values forms an arithmetic sequence.

To be sure you should always check the “outer points”.



Check the dy and dx on your table.

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