Up

Solving by substitution

Sometimes it is not possible or convenient to solve a system of equations by graphing.  In such a case we can turn to a method known as substitution to find the values of the variables.  To use the substitution method, we use the following procedure:

  1. Choose either of the two equations to begin with
  2. Solve for one of the variables in terms of the other
  3. Substitute the expression into the other equation
  4. You will obtain the value of one of the variables
  5. Substitute this value into either of the original equations to obtain the value of the other variable.

Let’s jump to an example.

Example:  Solve the system by substitution

\( - 8x + 5y =  - 6\)
\( - 3x + y =  - 4\)

Solution:  We follow the first procedure.  We want to choose either of the two equations to begin with.  We choose the second equation because it is easier to solve for the variable \(y\) in that equation.  Solving for \(y\) in the second equation gives us

\( - 3x + y =  - 4\)
\(y = 3x - 4\)

Continuing the procedure, we substitute the expression \(3x - 4\) for \(y\) in the other original equation.  We have

\( - 8x + 5y =  - 6\)
\( - 8x + 5\left( {3x - 4} \right) =  - 6\)
\( - 8x + 15x - 20 =  - 6\)
\(7x - 20 =  - 6\)
\(7x = 14\)
\(x = 2\)

We have obtained a value for one of the variables.  We substitute this value into either of the original equations.  We will substitute it into the second original equation:

\( - 3x + y =  - 4\)
\( - 3\left( 2 \right) + y =  - 4\)
\( - 6 + y =  - 4\)
\(y = 2\)

The solution is (2, 2)

Another Example:  Solve the system by substitution

\(7x + y =  - 15\)
\( - 6x - 7y =  - 24\)

Solution:  Here we choose to begin work with the first equation, solving for \(y\).  We have

\( - 6x - 7y =  - 24\)
\( - 6x - 7\left( { - 7x - 15} \right) =  - 24\)
\( - 6x + 49x + 105 =  - 24\)
\(43x + 105 =  - 24\)
\(43x =  - 129\)
\(x =  - 3\)

We substitute this value into either of the original equations.  We choose the first equation.  We have

\(7x + y =  - 15\)
\(7\left( { - 3} \right) + y =  - 15\)
\( - 21 + y =  - 15\)
\(y = 6\)

The solution is (-3, 6).

Below you can download some free math worksheets and practice.


Downloads:
11761 x

Solve each system by substitution.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Systems-of-Equations-and-Inequalities-Solving-by-substitution-easy

Watch below how to solve this example:

 

Downloads:
9627 x

Solve each system by substitution.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Systems-of-Equations-and-Inequalities-Solving-by-substitution-medium

Watch below how to solve this example:

 

Downloads:
9541 x

Solve each system by substitution.

This free worksheet contains 10 assignments each with 24 questions with answers.

Example of one question:

Systems-of-Equations-and-Inequalities-Solving-by-substitution-hard

Watch below how to solve this example:

 
 
 

Facebook PageYouTube Channel

Algebra and Pre-Algebra

Beginning Algebra
Adding and subtracting integer numbers
Dividing integer numbers
Multiplying integer numbers
Sets of numbers
Order of operations
The Distributive Property
Verbal expressions
Beginning Trigonometry
Finding angles
Finding missing sides of triangles
Finding sine, cosine, tangent
Equations
Absolute value equations
Distance, rate, time word problems
Mixture word problems
Work word problems
One step equations
Multi step equations
Exponents
Graphing exponential functions
Operations and scientific notation
Properties of exponents
Writing scientific notation
Factoring
By grouping
Common factor only
Special cases
Linear Equations and Inequalities
Plotting points
Slope
Graphing absolute value equations
Percents
Percent of change
Markup, discount, and tax
Polynomials
Adding and subtracting
Dividing
Multiplying
Naming
Quadratic Functions
Completing the square by finding the constant
Graphing
Solving equations by completing the square
Solving equations by factoring
Solving equations by taking square roots
Solving equations with The Quadratic Formula
Understanding the discriminant
Inequalities
Absolute value inequalities
Graphing Single Variable Inequalities
Radical Expressions
Adding and subtracting
Dividing
Equations
Multiplying
Simplifying single radicals
The Distance Formula
The Midpoint Formula
Rational Expressions
Adding and subtracting
Equations
Multiplying and dividing
Simplifying and excluded values
Systems of Equations and Inequalities
Graphing systems of inequalities
Solving by elimination
Solving by graphing
Solving by substitution
Word problems