Solving Equations With Decimal Coefficients

Solving Equations With Decimal Coefficients

Some equations have decimals in them. This kind of equation will occur when we solve problems dealing with money or percentages. But decimals can also be expressed as fractions. For example, \(0.3=\frac{3}{10}\) and \(0.17=\frac{17}{100}\). So, with an equation with decimals, we can use the same method we used to clear fractions—multiply both sides of the equation by the least common denominator.

Example

Solve: \(0.06x+0.02=0.25x-1.5\).

Solution

Look at the decimals and think of the equivalent fractions.

\(0.06=\frac{6}{100}\phantom{\rule{2em}{0ex}}0.02=\frac{2}{100}\phantom{\rule{2em}{0ex}}0.25=\frac{25}{100}\phantom{\rule{2em}{0ex}}1.5=1\frac{5}{10}\)

Notice, the LCD is 100.

By multiplying by the LCD, we will clear the decimals from the equation.

Multiply both sides by 100.
Distribute.
Multiply, and now we have no more decimals.
Collect the variables to the right.
Simplify.
Collect the constants to the left.
Simplify.
Divide by 19.
Simplify.
Check: Let \(x=8\).

The next example uses an equation that is typical of the money applications in the next tutorial. Notice that we distribute the decimal before we clear all the decimals.

Example

Solve: \(0.25x+0.05\left(x+3\right)=2.85\).

Solution

Distribute first.
Combine like terms.
To clear decimals, multiply by 100.
Distribute.
Subtract 15 from both sides.
Simplify.
Divide by 30.
Simplify.
Check it yourself by substituting \(x=9\) into the original equation.

Key Concepts

• Strategy to Solve an Equation with Fraction Coefficients
  1. Find the least common denominator of all the fractions in the equation.
  2. Multiply both sides of the equation by that LCD. This clears the fractions.
  3. Solve using the General Strategy for Solving Linear Equations.