Solving One-Step Multiplication & Division Equations with Decimals
Solving one-step multiplication and division equations with decimals uses the properties of equality: to solve ax = b (multiplication equation), divide both sides by a; to solve x/a = b (division equation), multiply both sides by a. For 0.4x = 2.8, divide both sides by 0.4 to get x = 7. This Grade 7 math skill from Saxon Math, Course 2 extends equation solving to decimal contexts, ensuring students can work with real-world equations involving prices, measurements, and rates that naturally involve decimal values.
Key Concepts
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.
Common Questions
How do I solve a multiplication equation with decimals?
Divide both sides by the coefficient. For 0.4x = 2.8, divide both sides by 0.4: x = 2.8 divided by 0.4 = 7.
How do I solve a division equation with decimals?
Multiply both sides by the divisor. For x/0.5 = 6, multiply both sides by 0.5: x = 6 times 0.5 = 3.
What is the Division Property of Equality?
The Division Property of Equality states that dividing both sides of an equation by the same non-zero number maintains equality. For ax = b, divide by a to get x = b/a.
What is the Multiplication Property of Equality?
The Multiplication Property of Equality states that multiplying both sides of an equation by the same number maintains equality. For x/a = b, multiply by a to get x = ab.
When do students solve multiplication and division equations with decimals?
This is a Grade 7 algebra skill. Saxon Math, Course 2 covers decimal equations in Chapter 7 as part of the equation-solving progression.
What are common mistakes with decimal equations?
Students sometimes compute the decimal multiplication or division incorrectly, or apply the wrong operation (multiplying when they should divide). Always undo the operation applied to the variable using its inverse.
How do decimal equations appear in real-world problems?
If a sale price of $15 is 0.75 of the original price, the equation 0.75p = 15 gives the original price. Decimal equations arise naturally in discount, tax, and rate problems.