Substitution Key Concepts
 The solution of a system of linear equations can be found algebraically by substitution.
 To solve a system of two equations in x and y by the substitution method, first solve one
of the equations for x or y. Next, substitute the new expression for x or y into the
remaining equation and solve. Finally, substitute the value for x or y back into one of the
original equations to determine the value of the other variable.
Unlike graphing techniques, algebraic solutions to systems of equations offer an exact solution.
This solution is an ordered pair that describes the point of intersection of the two equations.
To solve by substitution, follow a five step process.
1. Choose one of the equations and isolate one of its variables.
2. Substitute the expression from step 1 in place of the corresponding variable in the other
equation.
3. Solve the new equation.
4. Substitute the number found in step 3 into one of the other equations and solve.
5. Check the solution by substituting the ordered pair into both original equations.
When solving by substitution, be strategic - look for a variable that is easy to isolate to save time.
Elimination Key Concepts
 A system of two linear equations in two variables can be solved algebraically by
elimination. This method involves the addition or subtraction of the equations in order to
eliminate one variable. Then, substitution is used to find the value of the other variable.
 Multiplication can be used to rewrite a system of equations so that one variable can be
eliminated by addition or subtraction.
Unlike graphing techniques, algebraic solutions to systems of equations offer an exact solution.
This solution is an ordered pair that describes the point of intersection of the two equations. If
the coefficients of the same variable in both equations have the same value or the opposite
value, you can eliminate that variable in a two equation linear system by adding or subtracting
the equations. This will result in a new equation with only one unknown.
To solve by elimination, follow a six step process.
1. Express both equations in the form ax + by = c.
2. Choose a variable to eliminate. If necessary, multiply one or both of the equations to
obtain the same or opposite coefficients for that variable.
3. Add or subtract like terms in the equation to eliminate the chosen variable.
4. Solve the resulting equation for the remaining variable.
5. Determine the value of the other variable by substituting the solved value of the known
variable into one of the other equations.
6. Verify your solution in the original equations.
When solving by elimination, be strategic - look for coefficients that are the same or opposite, or
can be made that way easily.