Solve Equations with Fractions or Decimals

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 do not 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 without fractions. This process is called “clearing” the equation of fractions.

Let’s solve a similar equation, but this time use the method that eliminates the fractions.

How to Solve Equations with Fraction Coefficients

Try it.

Solve: \(\frac{1}{6}y-\frac{1}{3}=\frac{5}{6}\).

Solution

Notice in , 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: \(6=\frac{1}{2}v+\frac{2}{5}v-\frac{3}{4}v\).

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 LCD of all fractions in the equation.
The LCD is 20.
Multiply both sides of the equation by 20.
Distribute.
Simplify, notice, no more fractions!
Combine like terms.
Divide by 3.
Simplify.
Check:
Let \(v=40\).

In the next example, we again have variables on both sides of the equation.

Condensed: the full section is in OpenStax Elementary Algebra 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 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

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 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 chapter. Notice that we distribute the decimal before we clear all the decimals.

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 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.

Solve Equations with Fractions or Decimals

Solve Equations with Fraction Coefficients

In the following exercises, solve each equation with fraction coefficients.

Try it.

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

Try it.

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

Solution

\(x=1\)

Try it.

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

Try it.

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

Solution

\(y=-1\)

Try it.

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

Try it.

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

Solution

\(b=-2\)

Try it.

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

Try it.

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

Solution

\(x=3\)

Try it.

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

Try it.

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

Solution

\(n=-24\)

Try it.

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

Try it.

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

Solution

\(x=-4\)

Try it.

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

Try it.

\(\frac{3}{2}z+\frac{1}{3}=z-\frac{2}{3}\)

Solution

\(z=-2\)

Try it.

\(\frac{1}{2}x-\frac{1}{4}=\frac{1}{12}x+\frac{1}{6}\)

Try it.

\(\frac{1}{2}a-\frac{1}{4}=\frac{1}{6}a+\frac{1}{12}\)

Solution

\(a=1\)

Try it.

\(\frac{1}{3}b+\frac{1}{5}=\frac{2}{5}b-\frac{3}{5}\)

Try it.

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

Solution

\(x=-6\)

Try it.

\(1=\frac{1}{6}(12x-6)\)

Try it.

\(1=\frac{1}{5}(15x-10)\)

Solution

\(x=1\)

Try it.

\(\frac{1}{4}(p-7)=\frac{1}{3}(p+5)\)

Try it.

\(\frac{1}{5}(q+3)=\frac{1}{2}(q-3)\)

Solution

\(q=7\)

Try it.

\(\frac{1}{2}(x+4)=\frac{3}{4}\)

Try it.

\(\frac{1}{3}(x+5)=\frac{5}{6}\)

Solution

\(x=-\frac{5}{2}\)

Try it.

\(\frac{5q-8}{5}=\frac{2q}{10}\)

Try it.

\(\frac{4m+2}{6}=\frac{m}{3}\)

Solution

\(m=-1\)

Try it.

\(\frac{4n+8}{4}=\frac{n}{3}\)

Try it.

\(\frac{3p+6}{3}=\frac{p}{2}\)

Solution

\(p=-4\)

Try it.

\(\frac{u}{3}-4=\frac{u}{2}-3\)

Try it.

\(\frac{v}{10}+1=\frac{v}{4}-2\)

Solution

\(v=20\)

Try it.

\(\frac{c}{15}+1=\frac{c}{10}-1\)

Try it.

\(\frac{d}{6}+3=\frac{d}{8}+2\)

Solution

\(d=-24\)

Try it.

\(\frac{3x+4}{2}+1=\frac{5x+10}{8}\)

Try it.

\(\frac{10y-2}{3}+3=\frac{10y+1}{9}\)

Solution

\(y=-1\)

Try it.

\(\frac{7u-1}{4}-1=\frac{4u+8}{5}\)

Try it.

\(\frac{3v-6}{2}+5=\frac{11v-4}{5}\)

Solution

\(v=4\)

Solve Equations with Decimal Coefficients

In the following exercises, solve each equation with decimal coefficients.

Try it.

\(0.6y+3=9\)

Try it.

\(0.4y-4=2\)

Solution

\(y=15\)

Try it.

\(3.6j-2=5.2\)

Try it.

\(2.1k+3=7.2\)

Solution

\(k=2\)

Try it.

\(0.4x+0.6=0.5x-1.2\)

Try it.

\(0.7x+0.4=0.6x+2.4\)

Solution

\(x=20\)

Try it.

\(0.23x+1.47=0.37x-1.05\)

Try it.

\(0.48x+1.56=0.58x-0.64\)

Solution

\(x=22\)

Try it.

\(0.9x-1.25=0.75x+1.75\)

Try it.

\(1.2x-0.91=0.8x+2.29\)

Solution

\(x=8\)

Try it.

\(0.05n+0.10(n+8)=2.15\)

Try it.

\(0.05n+0.10(n+7)=3.55\)

Solution

\(n=19\)

Try it.

\(0.10d+0.25(d+5)=4.05\)

Try it.

\(0.10d+0.25(d+7)=5.25\)

Solution

\(d=10\)

Try it.

\(0.05(q-5)+0.25q=3.05\)

Try it.

\(0.05(q-8)+0.25q=4.10\)

Solution

\(q=15\)

Condensed: the full section is in OpenStax Elementary Algebra 2e.

Haddaad Ma jiro xisaabis oo ku nool midkan, laakiin qaybaha ka mid ah waa la xisaabi karaa. Ku raaxayso mid ka hooseeya, ama ku qor adiga kuu gaar ah.

⌨ Nooc ✎ Sawir

Su'aalkaaga xisaabta

π Xalin

Tababar (40)

Ku day mid kasta oo ka mid ah oo ku saabsan buugaag hore. Muuji jawaabta si aad u hubiso; kuwa la xaqiijiyay waxaa la furi karaa in xalka ah tallaabo kasta.

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

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   \(3\)

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

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   \(12\)

3. Multiply 4.78 by 100.

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   \(478\)

4. Solve: \(\frac{1}{6}y-\frac{1}{3}=\frac{5}{6}\).

   Shaqada iyada oo Tallaabo

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

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

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

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

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   \(x=-2\)

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

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   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 LCD of all fractions in the equation.
   The LCD is 20.
   Multiply both sides of the equation by 20.
   Distribute.
   Simplify, notice, no more fractions!
   Combine like terms.
   Divide by 3.
   Simplify.
   Check:
   Let \(v=40\).

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

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   \(x=12\)

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

   Shaqada iyada oo Tallaabo

   Jawaabta muuji

   \(u=-12\)

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    Find the LCD of all fractions in the equation. The LCD is 8.
    Multiply both sides by the LCD.
    Distribute.
    Simplify, no more fractions.
    Subtract \(3a\) from both sides.
    Simplify.
    Subtract 6 from both sides.
    Simplify.
    Divide by 5.
    Simplify.
    Check:
    Let \(a=-2\).

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(x=-1\)

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(c=-2\)

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    Distribute.
    Simplify. Now there are no fractions.
    Subtract 1 from both sides.
    Simplify.
    Divide by 2.
    Simplify.
    Check:
    Let \(x=-3\).

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(p=-4\)

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(q=2\)

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    Distribute.
    Simplify.
    Multiply by the LCD, 4.
    Distribute.
    Simplify.
    Collect the variables to the left.
    Simplify.
    Collect the constants to the right.
    Simplify.
    Check:
    Let \(y=9\).
    Finish the check on your own.

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(n=2\)

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(m=-1\)

19. Solve: \(\frac{5x-3}{4}=\frac{x}{2}\).

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    Multiply by the LCD, 4.
    Simplify.
    Collect the variables to the right.
    Simplify.
    Divide.
    Simplify.
    Check:
    Let \(x=1\).

20. Solve: \(\frac{4y-7}{3}=\frac{y}{6}\).

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(y=2\)

21. Solve: \(\frac{-2z-5}{4}=\frac{z}{8}\).

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(z=-2\)

22. Solve: \(\frac{a}{6}+2=\frac{a}{4}+3\).

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    Multiply by the LCD, 12.
    Distribute.
    Simplify.
    Collect the variables to the right.
    Simplify.
    Collect the constants to the left.
    Simplify.
    Check:
    Let \(a=-12\).

23. Solve: \(\frac{b}{10}+2=\frac{b}{4}+5\).

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(b=-20\)

24. Solve: \(\frac{c}{6}+3=\frac{c}{3}+4\).

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(c=-6\)

25. Solve: \(\frac{4q+3}{2}+6=\frac{3q+5}{4}\).

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    Multiply by the LCD, 4.
    Distribute.
    Simplify.
    Collect the variables to the left.
    Simplify.
    Collect the constants to the right.
    Simplify.
    Divide by 5.
    Simplify.
    Check:
    Let \(q=-5\).
    Finish the check on your own.

26. Solve: \(\frac{3r+5}{6}+1=\frac{4r+3}{3}\).

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(r=1\)

27. Solve: \(\frac{2s+3}{2}+1=\frac{3s+2}{4}\).

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(s=-8\)

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    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 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 \(x=8\).

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(h=12\)

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(k=-1\)

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    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.

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(n=9\)

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(d=16\)

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

    Shaqada iyada oo Tallaabo

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(x=1\)

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

    Shaqada iyada oo Tallaabo

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(y=-1\)

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

    Shaqada iyada oo Tallaabo

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

    Shaqada iyada oo Tallaabo

    Jawaabta muuji

    \(b=-2\)

40. \(2=\frac{1}{3}x-\frac{1}{2}x+\frac{2}{3}x\)

    Shaqada iyada oo Tallaabo

Ku hay shaqadaada

A xisaab bilaash ah oo ku daray qoraalada waxbarasho kasta, diiwaanka waxa aad dhamaatay, dhibaatooyinka aad xal u tahay meel kaliya, iyo macallinka aad weydiisan kartaa oo ku saabsan bogga. Xisaabta oo keliya waa u furan yahay qof kasta, la diiwaan geliyey ama aan.

Ku soo biir Ku soo gal

Calaamadaha la isticmaalay halkan

Taab calaamad kasta oo loogu talagalay qeexidda buuxda, sawir, iyo waxa xarfaha oo dhan ka mid ah ka dhigan tahay.

f'(x)derivative
Instantaneous rate of change; slope of the graph.\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.

Sida loo: Solve Equations with Fractions or Decimals

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 all the fractions in the equation.
7. Multiply both sides of the equation by that LCD. This clears the fractions.
8. Solve using the General Strategy for Solving Linear Equations.

Su'aalaha dadku waydiiyaan

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.

Qaybaha ee boggaan waxaa la beddelay ka OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed iyo dib-oo halkan lagu sharaxay; khaladaadka waa inoo. Sababtoo ah asalka waxaa loo oggolaaday in aan loo isticmaalin ganacsi, fasalkan waa bilaash, waxba ma iibiyo, iyo waxaa lagu wadaagayaa liisan la mid ah.

In ka badan Algebra

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