Mathematics » Solving Linear Equations I » Solve Equations with Fraction or Decimal Coefficients

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 and percent. But decimals are really another way to represent fractions. For example, \(0.3=\frac{3}{10}\) and \(0.17=\frac{17}{100}.\) So, when we have an equation with decimals, we can use the same process we used to clear fractions—multiply both sides of the equation by the least common denominator.

Example

Solve: \(0.8x-5=7.\)

Solution

The only decimal in the equation is \(0.8.\) Since \(0.8=\frac{8}{10},\) the LCD is \(10.\) We can multiply both sides by \(10\) to clear the decimal.

 Solving Equations with Decimal Coefficients
Multiply both sides by the LCD.Solving Equations with Decimal Coefficients
Distribute.Solving Equations with Decimal Coefficients
Multiply, and notice, no more decimals!Solving Equations with Decimal Coefficients
Add 50 to get all constants to the right.Solving Equations with Decimal Coefficients
Simplify.Solving Equations with Decimal Coefficients
Divide both sides by 8.Solving Equations with Decimal Coefficients
Simplify.Solving Equations with Decimal Coefficients
Check: Let \(x=15.\) 
Solving Equations with Decimal Coefficients 

Example

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

Solution

Look at the decimals and think of the equivalent fractions.

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

Notice, the LCD is \(100.\)

By multiplying by the LCD we will clear the decimals.

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

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

Example

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

Solution

 Solving Equations with Decimal Coefficients
Distribute first.Solving Equations with Decimal Coefficients
Combine like terms.Solving Equations with Decimal Coefficients
To clear decimals, multiply by 100.Solving Equations with Decimal Coefficients
Distribute.Solving Equations with Decimal Coefficients
Subtract 15 from both sides.Solving Equations with Decimal Coefficients
Simplify.Solving Equations with Decimal Coefficients
Divide by 30.Solving Equations with Decimal Coefficients
Simplify.Solving Equations with Decimal Coefficients
Check: Let \(x=9.\) 
Solving Equations with Decimal Coefficients 

Optional Video: Solving Multiple Step Equations Involving Decimals

Key Concepts

  • Solve equations with fraction coefficients by clearing the fractions.
    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.

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