Solve Equations with Fraction or Decimal Coefficients

Solve Equations with Fraction Coefficients

Let’s use the General Strategy for Solving Linear Equations introduced earlier to solve the equation \(\frac{1}{8}\ x+\frac{1}{2}=\frac{1}{4}.\)

To isolate the \(x\) term, subtract \(\frac{1}{2}\) from both sides.
Simplify the left side.
Change the constants to equivalent fractions with the LCD.
Subtract.
Multiply both sides by the reciprocal of \(\frac{1}{8}\).
Simplify.

This method worked fine, but many students don’t feel very confident when they see all those fractions. So we are going to show an alternate method to solve equations with fractions. This alternate method eliminates the fractions.

We will apply the Multiplication Property of Equality and multiply both sides of an equation by the least common denominator of all the fractions in the equation. The result of this operation will be a new equation, equivalent to the first, but with no fractions. This process is called clearing the equation of fractions. Let’s solve the same equation again, but this time use the method that clears the fractions.

Example

Try it.

Solve: \(\frac{1}{8}\ x+\frac{1}{2}=\frac{1}{4}.\)

Solution

Find the least common denominator of all the fractions in the equation.
Multiply both sides of the equation by that LCD, 8. This clears the fractions.
Use the Distributive Property.
Simplify, and notice, no more fractions!
Solve using the General Strategy for Solving Linear Equations.
Simplify.
Check: Let \(x=-2\)

Notice in that once we cleared the equation of fractions, the equation was like those we solved earlier in this chapter. We changed the problem to one we already knew how to solve! We then used the General Strategy for Solving Linear Equations.

Example

Try it.

Solve: \(7=\frac{1}{2}\ x+\frac{3}{4}\ x-\frac{2}{3}\ x.\)

Solution

We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation.

Find the least common denominator of all the fractions in the equation.
Multiply both sides of the equation by 12.
Distribute.
Simplify, and notice, no more fractions!
Combine like terms.
Divide by 7.
Simplify.
Check: Let \(x=12.\)

Condensed: the full section is in OpenStax Prealgebra 2e.

Solve 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

Try it.

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.

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

Example

Try it.

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

Solution

Look at the decimals and think of the equivalent fractions.

\(0.06=\frac{6}{100},\ 0.02=\frac{2}{100},\ 0.25=\frac{25}{100},\ 1.5=1\frac{5}{10}\)

Notice, the LCD is \(100.\)

By multiplying by the LCD we will clear the decimals.

Multiply both sides by 100.
Distribute.
Multiply, and now 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 ones we will see in the money applications in the next chapter. Notice that we will distribute the decimal first before we clear all decimals in the equation.

Example

Try it.

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

Solution

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

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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Pratica (40)

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1. Multiply: \(8\cdot \frac{3}{8}.\)

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   Rivela la risposta

   \(3\)

2. Find the LCD of \(\frac{5}{6}\ \text{and}\ \frac{1}{4}.\)

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   Rivela la risposta

   \(12\)

3. Multiply: \(4.78\) by \(100.\)

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   Rivela la risposta

   \(478\)

4. Solve: \(\frac{1}{8}\ x+\frac{1}{2}=\frac{1}{4}.\)

   Lavoraci sopra. Suggerimento

   Rivela la risposta

   Find the least common denominator of all the fractions in the equation.
   Multiply both sides of the equation by that LCD, 8. This clears the fractions.
   Use the Distributive Property.
   Simplify, and notice, no more fractions!
   Solve using the General Strategy for Solving Linear Equations.
   Simplify.
   Check: Let \(x=-2\)

5. Solve: \(\frac{1}{4}\ x+\frac{1}{2}=\frac{5}{8}.\)

   Lavoraci sopra. Suggerimento

   Rivela la risposta

   \(x=\frac{1}{2}\)

6. Solve: \(\frac{1}{6}\ y-\frac{1}{3}=\frac{1}{6}.\)

   Lavoraci sopra. Suggerimento

   Rivela la risposta

   y = 3

7. Solve: \(7=\frac{1}{2}\ x+\frac{3}{4}\ x-\frac{2}{3}\ x.\)

   Lavoraci sopra. Suggerimento

   Rivela la risposta

   We want to clear the fractions by multiplying both sides of the equation by the LCD of all the fractions in the equation.
   

   Find the least common denominator of all the fractions in the equation.
   Multiply both sides of the equation by 12.
   Distribute.
   Simplify, and notice, no more fractions!
   Combine like terms.
   Divide by 7.
   Simplify.
   Check: Let \(x=12.\)

8. Solve: \(6=\frac{1}{2}\ v+\frac{2}{5}\ v-\frac{3}{4}\ v.\)

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   Rivela la risposta

   v = 40

9. Solve: \(-1=\frac{1}{2}\ u+\frac{1}{4}\ u-\frac{2}{3}\ u.\)

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   Rivela la risposta

   u = −12

10. Solve: \(x+\frac{1}{3}=\frac{1}{6}\ x-\frac{1}{2}.\)

    Lavoraci sopra. Suggerimento

    Rivela la risposta

    Find the LCD of all the fractions in the equation.
    Multiply both sides by the LCD.
    Distribute.
    Simplify, no more fractions!
    Subtract \(x\) from both sides.
    Simplify.
    Subtract 2 from both sides.
    Simplify.
    Divide by 5.
    Simplify.
    Check: Substitute \(x=-1.\)

11. Solve: \(a+\frac{3}{4}=\frac{3}{8}\ a-\frac{1}{2}.\)

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    Rivela la risposta

    a = −2

12. Solve: \(c+\frac{3}{4}=\frac{1}{2}\ c-\frac{1}{4}.\)

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    Rivela la risposta

    c = −2

13. Solve: \(1=\frac{1}{2}(4x+2).\)

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    Rivela la risposta

    Distribute.
    Simplify. Now there are no fractions to clear!
    Subtract 1 from both sides.
    Simplify.
    Divide by 2.
    Simplify.
    Check: Let \(x=0.\)

14. Solve: \(-11=\frac{1}{2}(6p+2).\)

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    Rivela la risposta

    p = −4

15. Solve: \(8=\frac{1}{3}(9q+6).\)

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    Rivela la risposta

    q = 2

16. Solve: \(\frac{1}{2}(y-5)=\frac{1}{4}(y-1).\)

    Lavoraci sopra. Suggerimento

    Rivela la risposta

    Distribute.
    Simplify.
    Multiply by the LCD, 4.
    Distribute.
    Simplify.
    Collect the \(y\) terms to the left.
    Simplify.
    Collect the constants to the right.
    Simplify.
    Check: Substitute \(9\) for \(y.\)

17. Solve: \(\frac{1}{5}(n+3)=\frac{1}{4}(n+2).\)

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    Rivela la risposta

    n = 2

18. Solve: \(\frac{1}{2}(m-3)=\frac{1}{4}(m-7).\)

    Lavoraci sopra. Suggerimento

    Rivela la risposta

    m = −1

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

    Lavoraci sopra. Suggerimento

    Rivela la risposta

    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.

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

20. Solve: \(0.6x-1=11.\)

    Lavoraci sopra. Suggerimento

    Rivela la risposta

    x = 20

21. Solve: \(1.2x-3=9.\)

    Lavoraci sopra. Suggerimento

    Rivela la risposta

    x = 10

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

    Lavoraci sopra. Suggerimento

    Rivela la risposta

    Look at the decimals and think of the equivalent fractions.

    \(0.06=\frac{6}{100},\ 0.02=\frac{2}{100},\ 0.25=\frac{25}{100},\ 1.5=1\frac{5}{10}\)

    Notice, the LCD is \(100.\)

    By multiplying by the LCD we will clear the decimals.
    

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

23. Solve: \(0.14h+0.12=0.35h-2.4.\)

    Lavoraci sopra. Suggerimento

    Rivela la risposta

    h = 12

24. Solve: \(0.65k-0.1=0.4k-0.35.\)

    Lavoraci sopra. Suggerimento

    Rivela la risposta

    k = −1

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

    Lavoraci sopra. Suggerimento

    Rivela la risposta

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

26. Solve: \(0.25n+0.05(n+5)=2.95.\)

    Lavoraci sopra. Suggerimento

    Rivela la risposta

    n = 9

27. Solve: \(0.10d+0.05(d-5)=2.15.\)

    Lavoraci sopra. Suggerimento

    Rivela la risposta

    d = 16

28. \(\frac{1}{4}\ x-\frac{1}{2}=-\frac{3}{4}\)

    Lavoraci sopra. Suggerimento

    Rivela la risposta

    x = −1

29. \(\frac{3}{4}\ x-\frac{1}{2}=\frac{1}{4}\)

    Lavoraci sopra. Suggerimento

30. \(\frac{5}{6}\ y-\frac{2}{3}=-\frac{3}{2}\)

    Lavoraci sopra. Suggerimento

    Rivela la risposta

    y = −1

31. \(\frac{5}{6}\ y-\frac{1}{3}=-\frac{7}{6}\)

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32. \(\frac{1}{2}\ a+\frac{3}{8}=\frac{3}{4}\)

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    Rivela la risposta

    \(a=\frac{3}{4}\)

33. \(\frac{5}{8}\ b+\frac{1}{2}=-\frac{3}{4}\)

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34. \(2=\frac{1}{3}\ x-\frac{1}{2}\ x+\frac{2}{3}\ x\)

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    Rivela la risposta

    x = 4

35. \(2=\frac{3}{5}\ x-\frac{1}{3}\ x+\frac{2}{5}\ x\)

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36. \(\frac{1}{4}\ m-\frac{4}{5}\ m+\frac{1}{2}\ m=-1\)

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    Rivela la risposta

    m = 20

37. \(\frac{5}{6}\ n-\frac{1}{4}\ n-\frac{1}{2}\ n=-2\)

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38. \(x+\frac{1}{2}=\frac{2}{3}\ x-\frac{1}{2}\)

    Lavoraci sopra. Suggerimento

    Rivela la risposta

    x = −3

39. \(x+\frac{3}{4}=\frac{1}{2}\ x-\frac{5}{4}\)

    Lavoraci sopra. Suggerimento

40. \(\frac{1}{3}\ w+\frac{5}{4}=w-\frac{1}{4}\)

    Lavoraci sopra. Suggerimento

    Rivela la risposta

    \(w=\frac{9}{4}\)

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Simboli utilizzati qui

Toccare qualsiasi simbolo per la definizione completa, un'immagine, e che cosa ogni lettera in esso significa.

\pmplus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.\neqnot equal
The two sides are different.\leqless/greater than or equal
Inequalities that allow equality; < and > exclude it.\sqrt{x}square root, n-th root
The non-negative number whose square (n-th power) is x.|x|absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.iimaginary unit
i² = −1.\log_b xlogarithm, natural log
The exponent b must be raised to for x; ln uses base e.\mathbb{N}number sets
Naturals, integers, rationals, reals, complex numbers.

Come: Solve Equations with Fraction or Decimal Coefficients

1. Solve equations with fraction coefficients
2. Solve equations with decimal coefficients
3. Find the least common denominator of
4. Multiply both sides of the equation by that LCD. This clears the fractions.
5. Solve using the General Strategy for Solving Linear Equations.
6. Find the least common denominator of
7. Multiply both sides of the equation by that LCD. This clears the fractions.
8. Solve using the General Strategy for Solving Linear Equations.

Domande che la gente chiede

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

Parti di questa pagina sono adattate da OpenStax Prealgebra 2e (CC BY-NC-SA 4.0). Condensato e ri-spiegato qui; gli errori sono nostri.

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