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

Solving Equations with Variables and Constants on Both Sides

Solving Equations with Variables and Constants on Both Sides

The next example will be the first to have variables and constants on both sides of the equation. As we did before, we’ll collect the variable terms to one side and the constants to the other side.

Example

Solve: \(7x+5=6x+2.\)

Solution

Start by choosing which side will be the variable side and which side will be the constant side. The variable terms are \(7x\) and \(6x.\) Since \(7\) is greater than \(6,\) make the left side the variable side and so the right side will be the constant side.

 Solving Equations with Variables and Constants on Both Sides
Collect the variable terms to the left side by subtracting \(6x\) from both sides.Solving Equations with Variables and Constants on Both Sides
Simplify.Solving Equations with Variables and Constants on Both Sides
Now, collect the constants to the right side by subtracting 5 from both sides.Solving Equations with Variables and Constants on Both Sides
Simplify.Solving Equations with Variables and Constants on Both Sides
The solution is \(x=-3\). 
Check: Let \(x=-3\). 
Solving Equations with Variables and Constants on Both Sides 

We’ll summarize the steps we took so you can easily refer to them.

How to Solve an equation with variables and constants on both sides.

  1. Choose one side to be the variable side and then the other will be the constant side.
  2. Collect the variable terms to the variable side, using the Addition or Subtraction Property of Equality.
  3. Collect the constants to the other side, using the Addition or Subtraction Property of Equality.
  4. Make the coefficient of the variable \(1,\) using the Multiplication or Division Property of Equality.
  5. Check the solution by substituting it into the original equation.

It is a good idea to make the variable side the one in which the variable has the larger coefficient. This usually makes the arithmetic easier.

Example

Solve: \(6n-2=-3n+7.\)

Solution

We have \(6n\) on the left and \(-3n\) on the right. Since \(6>-3,\) make the left side the “variable” side.

 Solving Equations with Variables and Constants on Both Sides
We don’t want variables on the right side—add \(3n\) to both sides to leave only constants on the right.Solving Equations with Variables and Constants on Both Sides
Combine like terms.Solving Equations with Variables and Constants on Both Sides
We don’t want any constants on the left side, so add 2 to both sides.Solving Equations with Variables and Constants on Both Sides
Simplify.Solving Equations with Variables and Constants on Both Sides
The variable term is on the left and the constant term is on the right.

 

To get the coefficient of \(n\) to be one, divide both sides by 9.

Solving Equations with Variables and Constants on Both Sides
Simplify.Solving Equations with Variables and Constants on Both Sides
Check: Substitute 1 for \(n\). 
Solving Equations with Variables and Constants on Both Sides 

Example

Solve: \(2a-7=5a+8.\)

Solution

This equation has \(2a\) on the left and \(5a\) on the right. Since \(5>2,\) make the right side the variable side and the left side the constant side.

 Solving Equations with Variables and Constants on Both Sides
Subtract \(2a\) from both sides to remove the variable term from the left.Solving Equations with Variables and Constants on Both Sides
Combine like terms.Solving Equations with Variables and Constants on Both Sides
Subtract 8 from both sides to remove the constant from the right.Solving Equations with Variables and Constants on Both Sides
Simplify.Solving Equations with Variables and Constants on Both Sides
Divide both sides by 3 to make 1 the coefficient of \(a\).Solving Equations with Variables and Constants on Both Sides
Simplify.Solving Equations with Variables and Constants on Both Sides
Check: Let \(a=-5\). 
Solving Equations with Variables and Constants on Both Sides 

Note that we could have made the left side the variable side instead of the right side, but it would have led to a negative coefficient on the variable term. While we could work with the negative, there is less chance of error when working with positives. The strategy outlined above helps avoid the negatives!

To solve an equation with fractions, we still follow the same steps to get the solution.

Example

Solve: \(\frac{3}{2}\phantom{\rule{0.1em}{0ex}}x+5=\frac{1}{2}\phantom{\rule{0.1em}{0ex}}x-3.\)

Solution

Since \(\frac{3}{2}>\frac{1}{2},\) make the left side the variable side and the right side the constant side.

 Solving Equations with Variables and Constants on Both Sides
Subtract \(\frac{1}{2}x\) from both sides.Solving Equations with Variables and Constants on Both Sides
Combine like terms.Solving Equations with Variables and Constants on Both Sides
Subtract 5 from both sides.Solving Equations with Variables and Constants on Both Sides
Simplify.Solving Equations with Variables and Constants on Both Sides
Check: Let \(x=-8\). 
Solving Equations with Variables and Constants on Both Sides 

We follow the same steps when the equation has decimals, too.

Example

Solve: \(3.4x+4=1.6x-5.\)

Solution

Since \(3.4>1.6,\) make the left side the variable side and the right side the constant side.

 Solving Equations with Variables and Constants on Both Sides
Subtract \(1.6x\) from both sides.Solving Equations with Variables and Constants on Both Sides
Combine like terms.Solving Equations with Variables and Constants on Both Sides
Subtract 4 from both sides.Solving Equations with Variables and Constants on Both Sides
Simplify.Solving Equations with Variables and Constants on Both Sides
Use the Division Property of Equality.Solving Equations with Variables and Constants on Both Sides
Simplify.Solving Equations with Variables and Constants on Both Sides
Check: Let \(x=-5\). 
Solving Equations with Variables and Constants on Both Sides 

Optional Video: Solve an Equation with Variables and Parentheses on Both Sides

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This is a lesson from the tutorial, Solving Linear Equations I and you are encouraged to log in or register, so that you can track your progress.

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