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.
Steps to solving Systems of Equations by Substitution:
1. Isolate a variable in one of the equations. (Either y = or x =).
2. Substitute the isolated variable in the other equation.
3. This will result in an equation with one variable. Solve the equation.
4. Substitute the solution from step 3 into another equation to solve for the other variable.
5. Recommended: Check the solution.
3x + 2y = 2 (equation 1)
y + 8 = 3x (equation 2)
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
3x + 6x - 16 = 2
Step 4: Combine like terms
9x - 16 = 2
Step 5: Isolate variable x
9x = 18
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
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