Free Algebra
Tutorials!
Home
Miscellaneous Equations
Operations with Fractions
Undefined Rational Expressions
Inequalities
Writing Equations for Lines Using Sequences
Intersections of Lines and Conics
Graphing Linear Equations
Solving Equations with Log Terms and Other Terms
Quadratic Expresions - Complete Squares
Adding and Subtracting Fractions with Like Denominators
Multiplying a Fraction by a Whole Number
Solving Equations with Log Terms and Other Terms
Solving Quadratic Equations by Factoring
Locating the Solutions of the Quadratic Equation
Properties of Exponents
Solving Equations with Log Terms on Each Side
http:
Graphs of Trigonometric Functions
Estimating Products and Quotients of Mixed Numbers
Inequalities
The circle
Adding Polynomials
Adding Fractions with Unlike Denominators
Factoring Polynomials
Linear Equations
Powers of Ten
Straight Lines
Dividing With Fractions
Multiplication Property of Equality
Rationalizing Denominators
Multiplying And Dividing Fractions
Distance Between Points on a Number Line
Solving Proportions Using Cross Multiplication
Using the Quadratic Formula
Scientific Notation
Imaginary Numbers
Values of Symbols for Which Fractions are Undefined
Graphing Equations in Three Variables
Writing Fractions as Decimals
Solving an Equation with Two Radical Terms
Solving Linear Systems of Equations by Elimination
Factoring Trinomials
Positive Rational Exponents
Adding and Subtracting Fractions
Negative Integer Exponents
Rise and Run
Brackets
Multiplying Square Roots
Multiplying Polynomials
Solving Systems of Linear Inequalities
Multiplication Property of Radicals
A Quadratic within a Quadratic
Graphing a Linear Equation
Calculations with Hundreds and Thousands
Multiplication Property of Square and Cube  Roots
Solving Equations with One Log Term
The Cartesian Coordinate Plane
Equivalent Fractions
Adding and Subtracting Square Roots
Solving Systems of Equations
Exponent Laws
Solving Quadratic Equations
Factoring Trinomials
Solving a System of Three Linear Equations by Elimination
Factoring Expressions
Adding and Subtracting Fractions
The parabola
Computations with Scientific Notation
Quadratic Equations
Finding the Greatest Common Factor
Introduction to Fractions
Simplifying Radical Expressions Containing One Term
Polynomial Equations
Graphing and Intercepts
The Number Line
Adding and Subtracting Rational Expressions with Different Denominators
Scientific Notation vs Standard Notation
Powers
Factoring by Grouping
Extraneous Roots
Variables and Expressions
Linera Equations
Integers and Substitutions
Squares and Square Roots
Adding and Subtracting Rational Expressions with Different Denominators
Solving Linear Inequalities
Expansion of a Product of Binomials
Powers and Exponents
Finding The Greatest Common Factor
Quadratic Functions
The Intercepts of a Parabola
Solving Equations Containing Rational Expressions
http:
Subtracting Polynomials
Solving Equations
Adding Fractions with Unlike Denominators
Solving Systems of Equations by Substitution
Solving Equations
Product and Quotient of Functions
Try the Free Math Solver or Scroll down to Tutorials!

 

 

 

 

 

 

 

 
 
 
 
 
 
 
 
 

 

 

 
 
 
 
 
 
 
 
 

Please use this form if you would like
to have this math solver on your website,
free of charge.


Solving Systems of Equations by Substitution

Solving a system by graphing is certainly limited by the accuracy of the graph. If the lines intersect at a point whose coordinates are not integers, then it is difficult to determine those coordinates from the graph. The method of solving a system by substitution does not depend on a graph and is totally accurate. For substitution we replace a variable in one equation with an equivalent expression obtained from the other equation. Our intention in this substitution step is to eliminate a variable and to give us an equation involving only one variable.

 

Example 1

An independent system solved by substitution

Solve the system by substitution:

2x + 3y = 8

y + 2x = 6

Solution

We can easily solve y + 2x = 6 for y to get y = -2x + 6. Now replace y in the first equation by -2x + 6:

2x + 3y = 8  
2x + 3(-2x + 6) = 8 Substitute -2x + 6 for y.
2x - 6x + 18 = 8  
-4x = -10  
x  

To find y, we let in the equation y = -2x + 6:

The next step is to check and y = 1 in each equation. If and y = 1 in 2x + 3y = 8, we get

If and y = 1 in y + 2x = 6, we get

Because both of these equations are true, the solution set to the system is . The equations of this system are independent.

 

Example 2

An inconsistent system solved by substitution

Solve by substitution:

x - 2y = 3

2x - 4y = 7

Solution

Solve the first equation for x to get x - 2y = 3 . Substitute 2y + 3 for x in the second equation:

2x - 4y = 7
2(2y + 3) = 7
4y + 6 - 4y = 7
6 = 7

Because 6 = 7 is incorrect no matter what values are chosen for x and y, there is no solution to this system of equations. The equations are inconsistent. To check, we write each equation in slope-intercept form:

x - 2y = 3 2x - 4y = 7
-2y = -x + 3 -4y = -2x + 7
y y

The graphs of these equations are parallel lines with different y-intercepts. The solution set to the system is the empty set, Ø.

 

All Right Reserved. Copyright 2005-2024