Final Review: Solving equations
The steps listed below to solve equations are general steps. They will guarantee that you will solve
the equations but not necessarily in the easiest way. If you are sure what you do is correct, you may use
different methods to solve them.
• Allowed operations on equations
1. Simplify either side.
2. Add (or subtract) the same real number or algebraic expression on both sides.
3. Multiply (or divide by) the same real number or algebraic expression, which is not 0, on both
sides.
4. Switch sides.
5. Raise to the same power on both sides.
6. Take exponential with the same base on both sides.
7. Take logarithm with the same base on both sides.
Always check your solution(s) in the original equation to avoid mistake.
• Solving linear equations
1. Use Operation 1 and 2 to isolate the variable.
2. Use Operation 3 to kill the coefficient before the variable.
3. Check your solution in the original equation.
• Solving absolute value equations
|a| can only have two values without absolute value, where a is an algebraic expression.
When a > 0 or a = 0, |a| = a. When a < 0, |a| = −a.
So separate the original equation into two cases.
Don’t forget to check your solution.
• Solving systems of linear equations
By substitution:
1. Solve one of the equations for one variable in terms of the other variable(s).
2. Substitute the solution in step 1 into the other equation(s) and solve it.
3. Back-substitute the solution in step 2 into the solution you get in step 1.
4. Check your solution in the original equations.
By elimination:
1. Use Operation 3 to make the coefficients of one variable to be the opposite to each other.
2. Add the equations.
3. Solve the equation you get in step 2.
4. Back-substitute the solution in step 3 into either of the original equations and solve it.
5. Check your solution in the original equations.
Choose the method that you like more and ignore the other one.
• Solving polynomial equations
1. Use Operation 1, 2, 3, and 4 to rewrite the equation in the form
polynomial (in standard form) = 0
2. Factor the polynomial on the left hand side.
3. Use zero-factor property and solve all the equations.
4. Check your solution in the original equation.
• Solving quadratic equations
By factoring: the same steps as polynomial equations.
By completing the square.
By quadratic formula: x =
−b ±
√
b2 − 4ac
when the equation is ax2 + bx + c = 0.
2a
• Solving rational equations
1. Use Operation 1, 2, 3, and 4 to rewrite the equation in the form
numerator polynomial (in standard form)
=0
denominator polynomial (in standard form)
2. Solve the equation:
numerator polynomial = 0
3. Check your solution in the original equation.
• Solving radical equations
1. Use Operation 1, 2, 3 and 4 to isolate a radical.
√
2. Use Operation 5 to kill the radical by the property ( n x)n = x.
3. Repeat step 1 and 2 if there are still some radicals.
4. Solve the equation without radical.
5. Check your solution in the original equation.
• Solving exponential and logarithmic equations
1. Use Operation 1, 2, 3 and 4 to isolate exponential or condense all logarithms into a single one.
2. Use Operation 6 or 7 to get the variable off the exponent or the logarithm by the property
loga ax = x or aloga x = x.
3. Solve it for x.
4. Check your solution in the original equation.
• Solving linear inequalities
The steps are similar as solving a linear equation. But when applying Operation 3 with a negative
number or Operation 4, switch the direction of the symbol of inequality.
For compound inequality, separate it into two inequalities. The solution is the common part of
the solution of these two inequalities.