"How to Solve a Linear Equation in One Variable Involving Fractions"

Linear equations are equations involving fractions that can be written in the form ax/n + b = c, where a, b, n, and c are constants and x is the unknown variable. Solving a linear equation means finding the value of x that makes the equation true. Sometimes, linear equations may involve fractions, which can make them seem more complicated.

 However, there are some simple steps that can help you solve any linear equation involving fractions.

 

Step 1: Clear the fractions by multiplying both sides of the equation by the least common denominator (LCD) of all the fractions. The LCD is the smallest number that can be divided by all the denominators of the fractions. For example, if the equation is 2/3 x - 1/4 = 5/6, the LCD is 12, because 12 is the smallest number that can be divided by 3, 4 and 6. Multiplying both sides by 12 gives:

 

(12) * (2/3 x - 1/4) = (12) * (5/6) , Use the distributive property.

8x - 3 = 10

 

Step 2: Simplify the equation by combining like terms and applying the distributive property if needed. Like terms are terms that have the same variable and exponent. The distributive property states that a(b + c) = ab + ac. For example, if the equation is 8x - 3 = 10, there are no like terms to combine and no distributive property to apply, so the equation is already simplified.

 

Step 3: Isolate x by using inverse operations to undo the operations on both sides of the equation. Inverse operations are operations that undo each other, such as addition and subtraction, multiplication and division, exponentiation and logarithm. For example, if the equation is 8x - 3 = 10, we can add 3 to both sides to undo the subtraction of 3 from the left hand side:

 

8x - 3 + 3 = 10 + 3

8x = 13

 

Then we can divide both sides by 8 to undo the multiplication by 8:

 

(8x) / 8 = (13) / 8

x = 13/8

 

Step 4: Check your solution by plugging it back into the original equation and simplifying. If both sides of the equation are equal, then your solution is correct. For example, if x = 13/8, we can plug it into the original equation:

 

2/3 x - 1/4 = 5/6

2/3 (13/8) - 1/4 = 5/6

26/24 - 6/24 = 5/6

20/24 = 5/6

5/6 = 5/6

 

Both sides are equal, so x = 13/8 is correct.

 

Conclusion

 

Solving a linear equation in one variable involving fractions may seem daunting at first, but it can be done easily by following these four steps:

 

- Clear the fractions by multiplying both sides by the LCD

- Simplify the equation by combining like terms and applying the distributive property

- Isolate x by using inverse operations

- Check your solution by plugging it back into the original equation

 

By following these steps, you can solve any linear equation involving fractions with confidence and accuracy.