Solving a linear equation involving fractions is the same as solving regular, non-fractional linear equations. Here, we just need to eliminate the fraction early in the process.
This elimination of fractions is done using the least common denominator (LCD) method. Once the fractions are removed, solving follows the same steps as regular linear equations.
Here, we will learn the steps followed to solve a few of such linear equations.
Steps
Let us solve: ${\dfrac{2}{3}x-\dfrac{5}{4}=\dfrac{1}{6}}$.
Step 1: Finding the Least Common Denominator (LCD)
The Least Common Denominator (LCD) represents the smallest multiple shared by all denominators in the equation. Identifying the LCD helps us to eliminate fractions.
Here, the denominators are:
3, 4, and 6
The LCD of 3, 4, and 6 is 12
Step 2: Multiplying Every Term by the LCD
Multiplying the entire equation by 12 to eliminate the fractions, we get
${12\left( \dfrac{2}{3}x-\dfrac{5}{4}\right) =12\times \dfrac{1}{6}}$
⇒ 8x – 15 = 2
Step 3: Solving for x
Adding 15 to both sides, we get
⇒ 8x – 15 + 15 = 2 + 15
⇒ 8x = 17
Now, dividing both sides by 8, we get
⇒ x = ${\dfrac{17}{8}}$
Thus, the value of the variable x is ${\dfrac{17}{8}}$
Solved Examples
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Solve ${\dfrac{3}{5}x+2=7}$
Solution:
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Given, ${\dfrac{3}{5}x+2=7}$
Since LCD is 5
⇒ ${5\left( \dfrac{3}{5}x+2\right) =5\times 7}$
⇒ 3x + 10 = 35
⇒ 3x + 10 – 10 = 35 – 10
⇒ 3x = 25
⇒ x = ${\dfrac{25}{3}}$
Thus, the solution is x = ${\dfrac{25}{3}}$
Problem: Solving linear equations with fractions on BOTH SIDES
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Solve ${\dfrac{4}{9}x-\dfrac{5}{3}=\dfrac{1}{2}}$
Solution:
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Given, ${\dfrac{2}{9}x-\dfrac{5}{3}=\dfrac{1}{2}}$
Since LCD of 9, 3, and 2 is 18
⇒ ${18\left( \dfrac{2}{9}x-\dfrac{5}{3}\right) =18\times \dfrac{1}{2}}$
⇒ 4x – 30 = 9
⇒ 4x – 30 + 30 = 9 + 30
⇒ 4x = 39
⇒ x = ${\dfrac{39}{4}}$
Thus, the solution is x = ${\dfrac{39}{4}}$
