Solving Systems of Equations by Substitution



This algebra lesson introduces the technique of solving systems of equations by substitution.

In some word problems, we may need to translate the sentences into more than one equation. If we have two unknown variables then we would need at least two equations to solve the variable. In general, if we have n unknown variables then we would need at least n equations to solve the variable.

In the Substitution Method, we isolate one of the variables in one of the equations and substitute the results in the other equation. We usually try to choose the equation where the coefficient of a variable is 1 and isolate that variable. This is to avoid dealing with fractions whenever possible. If none of the variables has a coefficient of 1 then you may want to consider the Addition Method or Elimination Method.

Related Topics:
Worksheets to practice solving systems of equations

More Algebra Lessons

Example:

3x + 2y = 2      (equation 1)
y + 8 = 3x        (equation 2)

Solution:

Step 1: Try to choose the equation where the coefficient of a variable is 1.

Choose equation 2 and isolate variable y
y = 3x – 8         (equation 3)

Step 2: From equation 3, we know that y is the same as 3x – 8

We can then substitute the variable y in equation 1 with 3x – 8
3x + 2 (3x – 8) = 2

Step 3: Remove brackets using distributive property

3x + 6x – 16 = 2

Step 4: Combine like terms

9x – 16 = 2

Step 5: Isolate variable x

9x = 18
x=18/9=2

Step 6: Substitute x = 2 into equation 3 to get the value for y

y = 3 (2) – 8
y = 6 – 8 = – 2

Step 7: Check your answer with equation 1

3 (2) + 2 (–2) = 6 – 4 = 2

Answer: x = 2 and y = –2

Solving Using Substitution Method through a series of mathematical steps to teach students algebra
2x + 5y = 6
9y + 2x =22



The following video shows another example of of solving systems of equations by substitution.
y = 2x + 5      
3x ? y =  
?9





This video explains the steps to solve a linear system of equations using the substitution method.
x + 3y = 12
2x + y = 6



The following is an example of a system of equations that is solved using the substitution method.
2x + 3y = 13
?2x + y = ?9





Solving Linear Systems of Equations Using Substitution
Include an explanation of the graphs - one solution, no solution, infinite solutions
2x + 4y = 4
y = x ? 2

x + 3y = 6
2x + 6y = ? 12

2x ? 3y = 6
4x ? 6y = 12



This video provides an example of how to solve a system of linear equation using the substitution method.
x + 2y = ?20
y = 2x





You can use the Mathway widget below to practice Algebra (Systems of Equations) or other math topics. Try the given examples, or type in your own problem. Then click "Answer" to check your answer.



(Clicking "View Steps" on the answer screen will take you to the Mathway site, where you can register for a free ten-day trial of the software.)





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