FreeAlgebra Tutorials!
 Home 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 Powers 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 http: 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 Fractions 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 Adding Quadratic Functions Conjugates Factoring 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 Percents Arithmetics 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 Polynomials Laws of Exponents Multiplying Polynomials Vertical Line Test http: Solving Inequalities with Fractions and Parentheses http: Multiplying Polynomials Fractions Solving Quadratic and Polynomial Equations Extraneous Solutions Fractions

Try the Free Math Solver or Scroll down to Tutorials!

 Depdendent Variable

 Number of equations to solve: 23456789
 Equ. #1:
 Equ. #2:

 Equ. #3:

 Equ. #4:

 Equ. #5:

 Equ. #6:

 Equ. #7:

 Equ. #8:

 Equ. #9:

 Solve for:

 Dependent Variable

 Number of inequalities to solve: 23456789
 Ineq. #1:
 Ineq. #2:

 Ineq. #3:

 Ineq. #4:

 Ineq. #5:

 Ineq. #6:

 Ineq. #7:

 Ineq. #8:

 Ineq. #9:

 Solve for:

 Please use this form if you would like to have this math solver on your website, free of charge. Name: Email: Your Website: Msg:

# 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 and Adding two even integers will result in an even integer as the sum. Even Ã— Odd (or Even) = Even   Even Â± 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

CHECK:

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

CHECK:

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 Ã— Odd = Odd Odd Â± Odd = Even Even Ã— Odd (or Even) = Even Even Â± 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 â€œcommon differencesâ€ for both x and y. Use the value for dx to complete the table values for x. For complete table see below. Note adding Â± 3 to given values forms an arithmetic sequence. To be sure you should always check the â€œouter pointsâ€.

Check:

Check:

Check the dy and dx on your table.