Linear Equations

Equations with Fractions

Solve a linear equation containing a fractional coefficient by multiplying both sides by the LCD first.

Example

Clear denominators with the LCD, then solve.

highlighted = computed this step

Step 1 — Set up

Set up the expression.

x3+2=5\frac{x}{3} + 2 = 5
Start with the denominator 3.left sidex/3 + 2=right side5LCDmultiply by 3 to clear x/3--left constantright sidethen subtract 6--check5 - 2 = 3Start: x / 3 + 2 = 5

Step 2 — Multiply by LCD

Clear denominators: multiply both sides by LCD 3: 3(x/3+2) = 3 x 5.

3(x3+2)=3×5\hl{3} \left( \frac{x}{3} + 2 \right) = \hl{3} \times 5
Multiply both sides by 3 to clear the denominator.left sidex/3 + 2=right side5LCD3(x/3 + 2) = 3 x 5--left constantright sidethen subtract 6--check5 - 2 = 3Start: x / 3 + 2 = 5

Step 3 — Clear denominators

After multiplying by 3, the equation is x + 6 = 15.

x+6=15x+ \hl{6} = \hl{15}
Clear the fraction: x + 6 = 15.left sidex/3 + 2=right side5LCD3(x/3 + 2) = 3 x 5x + 6 = 153 x 2 = 63 x 5 = 15then subtract 6--check5 - 2 = 3Start: x / 3 + 2 = 5

Step 4 — Subtract constant

Subtract the constant: 15 - 6 = 9.

x=15−6=9x = 15 - 6 = \hl{9}
Subtract 6 from both sides.left sidex/3 + 2=right side5LCD3(x/3 + 2) = 3 x 5x + 6 = 153 x 2 = 63 x 5 = 1515 - 6 = 9--check5 - 2 = 3Start: x / 3 + 2 = 5

Step 5 — Result

Read the final result.

x=9x = 9
Result: x = 9.left sidex/3 + 2=right side5LCD3(x/3 + 2) = 3 x 5x + 6 = 153 x 2 = 63 x 5 = 1515 - 6 = 9x = 9check9/3 + 2 = 5Start: x / 3 + 2 = 5
with-fractions To solve x/k + c = d: multiply both sides by the LCD first, then solve the cleared equation.