How to Solve Equations with Variables on Both Sides

Step-by-Step Guide

1. 1

   Step 1: Apply the distributive property

   The distributive property states that a(b+c)=ab+ac{\displaystyle a(b+c)=ab+ac}.This rule allows you to cancel out parentheses by multiplying each term in the parentheses by the number outside the parentheses.For example, if your equation is 2(10−2x)=4(2x+2){\displaystyle 2(10-2x)=4(2x+2)}, use the distributive property to multiply the terms in parentheses by the number outside the parentheses:2(10−2x)=4(2x+2){\displaystyle 2(10-2x)=4(2x+2)}20−4x=8x+8{\displaystyle 20-4x=8x+8}
2. 2

   Step 2: if necessary.

   To cancel the variable, complete the opposite operation as stated in the equation.
   
   For example, if the term is subtracted in the equation, cancel it by adding.
   
   If the term is added in the equation, cancel it by subtracting.
   
   It is usually easiest to cancel the variable with the smaller coefficient.For example, in the equation 20−4x=8x+8{\displaystyle 20-4x=8x+8}, cancel the term −4x{\displaystyle
   -4x} by adding 4x{\displaystyle 4x}:20−4x+4x=8x+8{\displaystyle 20-4x+4x=8x+8}. , Whatever you do to one side of the equation, you must do to the other side as well.
   
   So if you add or subtract to cancel the variable on one side of the equation, you must add or subtract to the other side as well.For example, if you added 4x{\displaystyle 4x} on one side of the equation to cancel the variable, you must also add 4x{\displaystyle 4x} to the other side of the equation:20−4x+4x=8x+8+4x{\displaystyle 20-4x+4x=8x+8+4x} , You should now have the variable on one side of the equation.
   
   For example:20−4x+4x=8x+8+4x{\displaystyle 20-4x+4x=8x+8+4x}20=12x+8{\displaystyle 20=12x+8} , You want the variable term on one side, and the constant on the other side.
   
   To move the constant to one side, add or subtract from each side of the equation to cancel the term on one side.For example, to cancel the +8{\displaystyle +8} constant on the variable side, subtract 8 from both sides of the equation:20=12x+8{\displaystyle 20=12x+8}20−8=12x+8−8{\displaystyle 20-8=12x+8-8}12=12x{\displaystyle 12=12x} , To do this, perform the operation opposite from the one denoted in the equation.
   
   Usually this will mean dividing to cancel a coefficient being multiplied by a variable.Remember that whatever you do to one side of the equation, you must do to the other side of the equation as well.
   
   For example, to cancel out the coefficient 12 from the equation, you would divide each side of the equation by 12:12=12x{\displaystyle 12=12x}1212=12x12{\displaystyle {\frac {12}{12}}={\frac {12x}{12}}}1=x{\displaystyle 1=x} , To make sure your answer is correct, substitute your solution back into the original equation.
   
   If the equation is true, your answer is correct.
   
   For example, if 1=x{\displaystyle 1=x}, substitute 1 for the variable in the equation and calculate:2(10−2x)=4(2x+2){\displaystyle 2(10-2x)=4(2x+2)}2(10−2(1))=4(2(1)+2){\displaystyle 2(10-2(1))=4(2(1)+2)}2(10−2)=4(2+2){\displaystyle 2(10-2)=4(2+2)}20−4=8+8{\displaystyle 20-4=8+8}16=16{\displaystyle 16=16}
3. 3

   Step 3: Cancel the variable on one side of the equation.

4. 4

   Step 4: Keep the equation balanced.

5. 5

   Step 5: Simplify the equation by combining like terms.

6. 6

   Step 6: Move the constants to one side of the equation

7. 7

   Step 7: if necessary.

8. 8

   Step 8: Cancel the variable’s coefficient.

9. 9

   Step 9: Check your work.

Detailed Guide

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