Solving Linear Equations with Fractions

Solving Linear Equations

Linear Equations with Fractions

Linear equations are equations of the first degree, meaning the variable is only to the first power. When fractions are involved, it is often helpful to clear denominators to simplify the equation.

• Linear Equation: An equation that can be written in the form ax + b = c, where a, b, and c are constants.

• Clearing Fractions: Multiply both sides of the equation by the least common denominator (LCD) to eliminate fractions.

Example Problem

Solve and check the linear equation:

$\frac{x+1}{16} = \frac{1}{8} + \frac{x-3}{9}$

Step 1: Find the Least Common Denominator (LCD)

• The denominators are 16, 8, and 9.

• The LCD of 16, 8, and 9 is 144.

Step 2: Multiply Both Sides by the LCD

$144 \left(\frac{x+1}{16}\right) = 144 \left(\frac{1}{8}\right) + 144 \left(\frac{x-3}{9}\right)$

Simplify each term:

• $144 \div 16 = 9$

• $144 \div 8 = 18$

• $144 \div 9 = 16$

$9(x+1) = 18(1) + 16(x-3)$

Step 3: Distribute and Simplify

• $9x + 9 = 18 + 16x - 48$

• $9x + 9 = 16x - 30$

Step 4: Solve for x

• Subtract $9x$ from both sides: $9 = 7x - 30$

• Add $30$to both sides: $39 = 7x$

• Divide both sides by $7$:$x = \frac{39}{7}$

Step 5: Check the Solution

Substitute $x = \frac{39}{7}$ back into the original equation to verify the solution.

Summary Table: Steps for Solving Linear Equations with Fractions

Step	Description
1	Find the least common denominator (LCD) of all fractions.
2	Multiply both sides of the equation by the LCD to clear fractions.
3	Simplify and solve the resulting linear equation.
4	Check the solution by substituting back into the original equation.

Key Points

• Always check your solution in the original equation, especially when fractions are involved.

• If the variable cancels and you get a true statement (like $0=0$), the solution is all real numbers.

• If the variable cancels and you get a false statement (like $0=5$), there is no solution.

Final Answer

• The solution set is $\left\{\frac{39}{7}\right\}$.