Method 2: Multiplication Property of Equality
The Multiplication Property of Equality is a Grade 7 algebra technique in Big Ideas Math, Course 2 for solving equations containing fractions. When a variable is divided by a number (or multiplied by a fraction), multiply both sides of the equation by the reciprocal of that coefficient to isolate the variable. For example, x/4 = 3 becomes x = 12 after multiplying both sides by 4. For (2/3)x = 8, multiply both sides by 3/2 to get x = 12. This property maintains equality because whatever operation is applied to one side must be applied to the other, preserving the balance of the equation.
Key Concepts
A proportion is an equation of the form $\frac{a}{b} = \frac{c}{d}$, where $b \neq 0$, $d \neq 0$. The proportion is read "$a$ is to $b$ as $c$ is to $d$." Since a proportion is an equation with fractions, we can solve it by multiplying both sides of the equation by the LCD (least common denominator) to clear the fractions. This method uses the multiplication property of equality to eliminate denominators.
Common Questions
What is the Multiplication Property of Equality?
If you multiply both sides of an equation by the same non-zero number, the equation remains true. It is used to eliminate a coefficient or denominator from a variable.
How do you use the Multiplication Property to solve x/4 = 3?
Multiply both sides by 4: (4)(x/4) = (4)(3), giving x = 12.
How do you solve (2/3)x = 8 using this property?
Multiply both sides by the reciprocal of 2/3, which is 3/2: (3/2)(2/3)x = (3/2)(8), so x = 12.
Why is multiplying by the reciprocal the right move for fraction coefficients?
A fraction times its reciprocal equals 1, which isolates the variable. For coefficient 2/3, multiply by 3/2 to get 1·x = x.
When is this method preferred over the Division Property of Equality?
When the coefficient is a fraction, multiplying by the reciprocal is more direct than dividing, since dividing by a fraction means multiplying by its reciprocal anyway.
Does multiplying both sides by a negative number change the equation?
In equations (not inequalities), no—the equation stays balanced. Unlike inequalities, there is no sign reversal when multiplying by a negative in an equation.