Overview:
Some algebra equations can be solved by combining operations. For example, students can use the Distributive Property with Multiplication over Addition to solve many different equations when the variables are the same. Combining operations can lead to shortcuts to solve problems.
What Is the Distributive Property?
The Distributive Property of Multiplication over Addition (or its sibling, the Distributive Property of Multiplication over Subtraction) is stated in algebra as true for any numbers a, b, and x, so that ax + bx = (a + b)x. For example, 2m + 3m equals (2 + 3)m or 5m. Simplifying 5y – 3y equals (5 – 3)y or 2y, showing that it works for subtraction, also.
Using an Alternate Form
Another form of the Distributive Property can be stated in algebraic terms as x(a + b) = xa + xb. This form of the equation works the same way because of the commutative property. For example, suppose you have two rectangles that are x units wide. One rectangle is 20 units long, and the other rectangle is 85 units long. The total area can be found by using the alternate form, as x(85 + 20) = x85 + x20 or x105.
Shortcuts for Mental Math
The Distributive Property can be used in a shortcut to do math mentally. For example, suppose one wants to leave a 20 percent tip on a meal that costs 25 dollars. A 10% tip on a meal that costs 25 dollars would be $2.50 and a 20% tip would be 2 times 10% or 2 times 2.50 or 5.00. Add 5 dollars to $25.00 and the total cost of the meal would be 30 dollars.
Equations in the Pattern ax + b = cx + d
This form of equations can be solved by combining operations. Suppose the problem were 3x + 5 = 10x + 26. The variable is on both sides of the equation, but operations can be combined so that all the variables will be on one side and all the constants will be on the other. In the next step, 3x – 10x + 5 = 10x – 10x + 26, or -7x + 5 = 26. Then -7x = 26 – 5, or -7x = 21 or -7x/-7 = 21/-7 or x = -3. To check, 3(-3) + 5 = -4 and 10(-3) + 26 = -30 + 26 = -4.
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