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

 Solving Equations Using a General Strategy
Simplify each side of the equation as much as possible.

 

Use the Distributive Property.

Solving Equations Using a General Strategy
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

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

Example

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

Solution

 Solving Equations Using a General Strategy
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.

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

 

Solving Equations Using a General Strategy

 

Solving Equations Using a General Strategy

 

Solving Equations Using a General Strategy

 
 

Example

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

Solution

 Solving Equations Using a General Strategy
Simplify each side of the equation as much as possible.

 

Distribute.

Solving Equations Using a General Strategy
Combine like termsSolving Equations Using a General Strategy
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.Solving Equations Using a General Strategy
Simplify.Solving Equations Using a General Strategy
Make the coefficient of the variable term equal to 1 by dividing both sides by 4.Solving Equations Using a General Strategy
Simplify.Solving Equations Using a General Strategy
Check: Let \(x=0\). 
Solving Equations Using a General Strategy 

Example

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

Solution

Be careful when distributing the negative.

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

Example

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

Solution

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

Example

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

Solution

 Solving Equations Using a General Strategy
Distribute.Solving Equations Using a General Strategy
Add \(x\) to get all the variables on the left.Solving Equations Using a General Strategy
Simplify.Solving Equations Using a General Strategy
Add 1 to get constants on the right.Solving Equations Using a General Strategy
Simplify.Solving Equations Using a General Strategy
Divide by 4.Solving Equations Using a General Strategy
Simplify.Solving Equations Using a General Strategy
Check: Let \(x=\frac{3}{2}\). 
Solving Equations Using a General Strategy 

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

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

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