Solving equations with parentheses

 
 
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How to solve equations in general

If an equation you need to solve has parentheses, simplify the parentheses (most often using distribution) and then solve as you normally would.

Solving Equations

  1. Simplify both sides of the equation as much as possible using the order of operations (distribute, combine like terms, etc.).

  2. If the variable (letter) you’re trying to solve for appears on both sides of the equation move one to the other side. Get all your “???x???’s to Texas” or get all the ???x???’s to one side of the equation.

  3. Solve by working backwards from the order of operations. Use inverse operations until the variable is alone and remember to do the same thing to both sides of the equation so that it stays balanced.

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Three examples of parentheses in equations that you can solve using distribution


 
 
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Distributing coefficients to solve the equation with parentheses on both sides

Example

Solve for the variable.

???-(3x+4)=6(2x-7)+8???


 

Start by simplifying both sides of the equation. You’ll need to distribute the coefficients in front of the parentheses.

???-3x-4=12x-42+8???

Simplify the right side of the equation by combining like terms.

???-3x-4=12x-34???

Move all of the ???x??? terms to one side of the equation by adding ???3x??? to both sides.

???-3x+3x-4=12x+3x-34???

???-4=15x-34???

Undo ???-34??? by adding ???34??? to both sides.

???-4+34=15x-34+34???

???30=15x???

Undo the multiplication of ???15??? by dividing both sides by ???15???.

???\frac{30}{15}=\frac{15x}{15}???

???2=x???

parentheses inside equations

To solve an equation that has parentheses, distribute to get rid of the parentheses, then follow order of operations to simplify the rest of the equation


Let’s try another example of solving equations with parentheses.


Another example of how to simplify equations with parentheses

Example

Solve for the variable, remembering that anything raised to the power of ???0??? is equal to ???1???.

???-(b-2^0)=7^0(b^0-4)-4(b+8)???


 

Start by simplifying anything raised to the power of ???0???. Remember that any nonzero value raised to the ???0??? is ???1???.

???-(b-1)=1(1-4)-4(b+8)???

Simplify the both sides of the equation by distributing.

???-b+1=1-4-4b-32???

Combine like terms on the right side (???1-4-32???).

???-b+1=-35-4b???

Move all of the ???b??? terms to one side of the equation by adding ???4b??? to both sides.

???-b+4b+1=-35-4b+4b???

???3b+1=-35???

Undo ???+1??? by subtracting ???1??? from both sides.

???3b+1-1=-35-1???

???3b=-36???

Undo the multiplication of ???3??? by dividing both sides by ???3???.

???\frac{3b}{3}=\frac{-36}{3}???

???b=-12???

 
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