Mathematics » Solving Linear Equations I » Solve Equations with Variables and Constants on Both Sides

Solving Equations Using a General Strategy

Solving Equations Using a General Strategy

Each of the first few sections of this tutorial has dealt with solving one specific form of a linear equation. It’s time now to lay out an overall strategy that can be used to solve any linear equation. We call this the general strategy. Some equations won’t require all the steps to solve, but many will. Simplifying each side of the equation as much as possible first makes the rest of the steps easier.

How to Use a general strategy for solving linear equations.

  1. Simplify each side of the equation as much as possible. Use the Distributive Property to remove any parentheses. Combine like terms.
  2. Collect all the variable terms to one side of the equation. Use the Addition or Subtraction Property of Equality.
  3. Collect all the constant terms to the other side of the equation. Use the Addition or Subtraction Property of Equality.
  4. Make the coefficient of the variable term to equal to \(1.\) Use the Multiplication or Division Property of Equality. State the solution to the equation.
  5. Check the solution. Substitute the solution into the original equation to make sure the result is a true statement.

Example

Solve: \(3\left(x+2\right)=18.\)

Solution

 .
Simplify each side of the equation as much as possible.

 

Use the Distributive Property.

.
Collect all variable terms on one side of the equation—all \(x\)s are already on the left side. 
Collect constant terms on the other side of the equation.

 

Subtract 6 from each side

.
Simplify..
Make the coefficient of the variable term equal to 1. Divide each side by 3..
Simplify..
Check: Let \(x=4\). 
. 

Example

Solve: \(-\left(x+5\right)=7.\)

Solution

 .
Simplify each side of the equation as much as possible by distributing.

 

The only \(x\) term is on the left side, so all variable terms are on the left side of the equation.

.
Add 5 to both sides to get all constant terms on the right side of the equation..
Simplify..
Make the coefficient of the variable term equal to 1 by multiplying both sides by -1..
Simplify..
Check: Let \(x=-12\). 
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Example

Solve: \(4\left(x-2\right)+5=-3.\)

Solution

 .
Simplify each side of the equation as much as possible.

 

Distribute.

.
Combine like terms.
The only \(x\) is on the left side, so all variable terms are on one side of the equation. 
Add 3 to both sides to get all constant terms on the other side of the equation..
Simplify..
Make the coefficient of the variable term equal to 1 by dividing both sides by 4..
Simplify..
Check: Let \(x=0\). 
. 

Example

Solve: \(8-2\left(3y+5\right)=0.\)

Solution

Be careful when distributing the negative.

 .
Simplify—use the Distributive Property..
Combine like terms..
Add 2 to both sides to collect constants on the right..
Simplify..
Divide both sides by −6..
Simplify..
Check: Let \(y=-\frac{1}{3}\). 
. 

Example

Solve: \(3\left(x-2\right)-5=4\left(2x+1\right)+5.\)

Solution

 .
Distribute..
Combine like terms..
Subtract \(3x\) to get all the variables on the right since \(8>3\)..
Simplify..
Subtract 9 to get the constants on the left..
Simplify..
Divide by 5..
Simplify..
Check: Substitute: \(-4=x\). 
. 

Example

Solve: \(\frac{1}{2}\left(6x-2\right)=5-x.\)

Solution

 .
Distribute..
Add \(x\) to get all the variables on the left..
Simplify..
Add 1 to get constants on the right..
Simplify..
Divide by 4..
Simplify..
Check: Let \(x=\frac{3}{2}\). 
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In many applications, we will have to solve equations with decimals. The same general strategy will work for these equations.

Example

Solve: \(0.24\left(100x+5\right)=0.4\left(30x+15\right).\)

Solution

 .
Distribute..
Subtract \(12x\) to get all the \(x\)s to the left..
Simplify..
Subtract 1.2 to get the constants to the right..
Simplify..
Divide..
Simplify..
Check: Let \(x=0.4\). 
. 

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