Lesson 5: Write and Solve Equations with Rational Numbers

Property

To solve for a variable, use inverse operations.

• Addition Property of Equality: If $a = b$, then $a + c = b + c$.
• Subtraction Property of Equality: If $a = b$, then $a - c = b - c$.

Examples

• Solve for $x$: $x + \frac{1}{5} = 2 \tfrac{4}{5}$

• Solve for $y$: $y - \frac{1}{3} = 1 \tfrac{1}{2}$

Property

To solve one-step addition or subtraction equations, use the inverse operation to isolate the variable. For an equation like $x + a = b$, subtract $a$ from both sides: $x = b - a$. For an equation like $x - a = b$, add $a$ to both sides: $x = b + a$.

Examples

• Solve for $x$: $x + 3.5 = 8.1$

• Solve for $y$: $y - 2.25 = 5.7$

Explanation

To solve a one-step equation involving addition or subtraction with decimals, you must isolate the variable. This is achieved by applying the inverse operation to both sides of the equation to maintain balance. If a number is added to the variable, subtract that number from both sides. If a number is subtracted from the variable, add that number to both sides.

Property

To solve a multiplication equation like $ax = b$, use the Division Property of Equality by dividing both sides by $a$: $x = \frac{b}{a}$. To solve a division equation like $\frac{x}{a} = b$, use the Multiplication Property of Equality by multiplying both sides by $a$: $x = ab$.

Examples

• Solve for $x$: $4.5x = 13.5$

To solve, divide both sides by $4.5$: $x = \frac{13.5}{4.5}$, so $x = 3$.

• Solve for $y$: $\frac{y}{2.1} = 5$

To solve, multiply both sides by $2.1$: $y = 5 \times 2.1$, so $y = 10.5$.

Explanation

This skill focuses on solving one-step equations that involve multiplying or dividing by a decimal number. To isolate the variable in a multiplication equation, you perform the inverse operation, which is division. Similarly, to solve a division equation, you multiply both sides by the decimal number. Remember to keep the equation balanced by applying the same operation to both sides.

Property

To solve a multiplication equation, multiply both sides by the reciprocal of the coefficient.
To solve a division equation, multiply both sides by the divisor.
Always convert mixed numbers to improper fractions before solving.

Examples

• Solve for $x$: $\frac{2}{3}x = \frac{5}{6}$

Multiply by the reciprocal of $\frac{2}{3}$, which is $\frac{3}{2}$:

• Solve for $y$: $y \div 1\frac{1}{4} = \frac{2}{5}$

Convert $1\frac{1}{4}$ to $\frac{5}{4}$. The equation is $y \div \frac{5}{4} = \frac{2}{5}$. Multiply both sides by $\frac{5}{4}$:

Explanation

This skill focuses on solving one-step equations involving multiplication or division with fractions. To isolate the variable in a multiplication equation, you use the inverse operation by multiplying both sides by the reciprocal of the fractional coefficient. For division equations, you multiply both sides by the fractional divisor to find the variable''s value. Remember to convert any mixed numbers into improper fractions as the first step to simplify the calculation.