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In Algebra a variable is an unknown quantity written as a letter.

 In order to solve for the unknown variable, you must isolate the variable. In order to isolate the variable you use the reverse of the order of operations ( PEMDAS.) Why reverse PEMDAS or SADMEP? In order to isolate the variable you have to undo the operations done to the variable. 

​When solving for a variable remember these three rules.

  • Isolate the variable
  • Whatever operation you perform on one side of the equation must be performed on the opposite of the equation.
  • Your order of operations becomes SADMEP

S=Subtract A=Addition D=Division M=Multiplication E= Exponent P=Parentheses

Let's apply these rules in an example involving a perimeter formula.

P = 2l + 2w and we want to solve for  l

Step 1. Isolate the variable l by subtracting  2w from each side.

             P -2w = 2l +2w

Step 2. P -2w = 2l   

Now we continue to isolate for l by dividing by 2 (Remember the same task must be performed to both sides)

p - 2w/2 = 2l/2

P-w = l   Solution

Notice in the example above we, 
  • Isolated the variable
  • Performed the same operation to both sides
  • Used SADMEP

video tutorial solving for a variable
video solution 1/x = 6/5x + 1

Today we are going to take an example of solving for a variable. We have half, base times height equals area. Hopefully you recognized this as the area of a triangle. We are going to solve for B or solve for the base. Now the steps to solve for a variable are as follows. First we are going to underline the variable that we are solving for. We are going to underline B and then use the reverse of please excuse my dear aunt sally. We are going to add and subtract first, multiple and divide next, and then any exponents then any parenthesis. So let’s work through our steps. OK let’s first underline the B ok that is what we are solving for. We next have multiplication and multiplication so we have to undo two multiplications and we will have to divide. Now if you are comfortable with multiplying with reciprocals that is the easiest way to get rid of a coefficient fraction. I’m going to multiply each side by two over one so I’m just going to multiple by the reciprocal of one half. So I end up with two over two so they just cancel out and if I multiply the left side by two then I have to multiply the right side by two over one which is just two So what I’m I left with after I multiply by the reciprocal? I’m left with base times height is equal to A times two. I like to put my coefficient first so I’m going to put a two times A. Now we have to get rid of the H because we are trying to get the B by itself. This is multiplication. How do we get rid of multiplication? We divide. Divide both sides by H and now we have the B by itself. So B is equal to two A over H and there is your answer.

Solve for Variable

Solving for a Variable

Step 2. Divide by 2
Step 3. Exponents, Square Root  (remember there are two roots for a square root)
Step 1. Add 5 to each side in order to isolate the variable
Example 2  Solve for m
Example 3 Solve for r         Solving for a variable with an exponent
Step 1. Subtract b from each side to isolate the variable.
Step 2. Divide by x
Step 3. Solution
Step 1. There isn't any  addition or subtraction so start by dividing  by x
Step 2. Square root in order to eliminate the exponent
Step 3. Solution
4pir^2 =A

1. √2x + 1 = 3^2

Solution = x = 4
2. 10 + √10x-1 = 13

Solution = x = 1
1. 4| n + 8 | = 56

Solution = n = 6   and n = -22

2. 3|4x -1 | -5 = 10

Solution = x = 3/2  and x = -1
1. x^2 = 110 -x

Solution = x = 10  x = -11

2. 2x^2 - 7x - 15 = 0

Solution = x = - 3/2  x =5
Solution = - 1/5 = x
Solution = No solution
video solution 4|n + 8| =56
Video Solution to both problems
video solution square root 2x + 1
Video Solution to both problems
video solution x^2 = 11-x
Video Solution to both problems
Video Solution to both problems
Example 1

Equations containing radicals - Solve for x in the following equations.

Equations containing absolute values.

Quadratic equations - Solve for x

Equations in involving rationals - Solve for x in the following equations.

How to solve for a variable with an exponent

Although the variable has an exponent you still follow the same steps as the first problem solved.