Lesson 3.2 – Solving Systems of Equations Algebraically
Because solving a system of equations graphically is not very accurate, we usually solve them algebraically.
Substitution Method
Steps for the substitution method
Solve one equation for one of the variables. (hint: if possible, choose the variable with a
or a
as
the coefficient)
Then, this expression is substituted for the variable in the other equation.
Once you find the value of a variable, substitute it back into either of the original equations to solve
for the remaining variable.
This ordered pair is the solution to the system and would be the point of intersection if we were to graph
both equations.
Use substitution to solve the system of equations.
Matt stopped for gas twice on a long car trip. The price of gas at the first station was
price at the second station was
spent
per gallon. Matt bought a total of
. How many gallons of gas did he buy at each station?
Define the variables:
Let
Let
Write the two equations and solve using the substitution method.
per gallon and the
gallons of gas and
Elimination Method
Add the two equations together to eliminate one of the variables
Solve for the remaining variable
Substitute back into one of the original equations to find the other variable
Ex.
Use the elimination method to solve the system of equations.
Sometimes adding the two equations will not eliminate either variable. You may use multiplication to write an
equivalent equation so that one of the variables has opposite coefficients in both equations.
Ex.
Solve using elimination. Hint: Begin with equations in standard form
Inconsistent System – when you add a system and the result is an equation that is never true.
Dependent System – when you add a system and the result is an equation that is always true.
Solve. Hint: Multiply both sides of the equation by the LCD to eliminate fractions.