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. It may take several steps to solve this equation, so we need a clear and organized strategy.

Example: How to Solve Equations with Variables and Constants on Both Sides

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

Solution

We’ll list the steps below so you can easily refer to them. But we’ll call this the ‘Beginning Strategy’ because we’ll be adding some steps later in this tutorial.

In Step 1, a helpful approach is to make the “variable” side the side that has the variable with the larger coefficient. This usually makes the arithmetic easier.

Example

Solve: \(8n-4=-2n+6.\)

Solution

In the first step, choose the variable side by comparing the coefficients of the variables on each side.

Since \(8>-2\), make the left side the “variable” side.
We don’t want variable terms on the right side—add \(2n\) to both sides to leave only constants on the right.
Combine like terms.
We don’t want any constants on the left side, so add \(4\) to both sides.
Simplify.
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 10.
Simplify.
Check:
Let \(n=1\).

Example

Solve: \(7a-3=13a+7.\)

Solution

In the first step, choose the variable side by comparing the coefficients of the variables on each side.

Since \(13>7\), make the right side the “variable” side and the left side the “constant” side.

Subtract \(7a\) from both sides to remove the variable term from the left.
Combine like terms.
Subtract \(7\) from both sides to remove the constant from the right.
Simplify.
Divide both sides by \(6\) to make \(1\) the coefficient of \(a\).
Simplify.
Check:
Let \(a=-\frac{5}{3}\).

In the last example, we could have made the left side the “variable” side, but it would have led to a negative coefficient on the variable term. (Try it!) While we could work with the negative, there is less chance of errors when working with positives. The strategy outlined above helps avoid the negatives!

To solve an equation with fractions, we just follow the steps of our strategy to get the solution!

Example

Solve: \(\frac{5}{4}x+6=\frac{1}{4}x-2.\)

Solution

Since \(\frac{5}{4}>\frac{1}{4}\), make the left side the “variable” side and the right side the “constant” side.

Subtract \(\frac{1}{4}x\) from both sides.
Combine like terms.
Subtract \(6\) from both sides.
Simplify.
\(\begin{array}{cccccc}\text{Check:}\hfill & & & \hfill \frac{5}{4}x+6& =\hfill & \frac{1}{4}x-2\hfill \\ \text{Let}\phantom{\rule{0.2em}{0ex}}x=-8.\hfill & & & \hfill \frac{5}{4}\left(-8\right)+6& \stackrel{?}{=}\hfill & \frac{1}{4}\left(-8\right)-2\hfill \\ & & & \hfill -10+6& \stackrel{?}{=}\hfill & -2-2\hfill \\ & & & \hfill -4& =\hfill & -4✓\hfill \end{array}\)

We will use the same strategy to find the solution for an equation with decimals.

Example

Solve: \(7.8x+4=5.4x-8.\)

Solution

Since \(7.8>5.4\), make the left side the “variable” side and the right side the “constant” side.

Subtract \(5.4x\) from both sides.
Combine like terms.
Subtract \(4\) from both sides.
Simplify.
Use the Division Propery of Equality.
Simplify.
Check:
Let \(x=-5\).

Key Concepts

• Beginning Strategy for Solving an Equation with Variables and Constants on Both Sides of the Equation
  1. Choose which side will be the “variable” side—the other side will be the “constant” side.
  2. Collect the variable terms to the “variable” side of the equation, using the Addition or Subtraction Property of Equality.
  3. Collect all the constants to the other side of the equation, using the Addition or Subtraction Property of Equality.
  4. Make the coefficient of the variable equal 1, using the Multiplication or Division Property of Equality.
  5. Check the solution by substituting it into the original equation.