Solving Multi-Step Linear Equations with Fractions

We need more than two operations to solve a linear equation . Use inverse operations to undo each operation in reverse order.

If an equation contains fractions, multiply both sides of the equation by the least common denominator (LCD) to clear fractions.

Steps for solving a Multi-Step Equation:

Step 1 Clear the equation of fractions.

Step 2 Use the Distributive Property to remove parentheses on each side.

Step 3 Combining like terms on each side.

Step 4 Undo addition or subtraction.

Step 5 Undo multiplication or division.

Example:

Solve $\frac{2 y}{3} + \frac{y}{2} = 7$ .

Solution

The least common denominator (LCD) in this case is $6$ . So, multiply both sides of the equation by $6$ .

$6 (\frac{2 y}{3} + \frac{y}{2}) = 6 (7)$

Use the distributive law on the left side of the equation.

$6 (\frac{2 y}{3}) + 6 (\frac{y}{2}) = 6 (7)$

Multiply.

$4 y + 3 y = 42$

Combine the like terms.

$7 y = 42$

Undo multiplication. Divide each side by $7$ .

$\frac{7 y}{7} = \frac{42}{7}$

Simplify.

$y = 6$