The distributive property states that the product of a number by a sum is equal to the sum of the individual products of the number for each of the addends. This means that a (b + c) = ab + ac. You can use this fundamental property to solve and simplify various types of equations. If you want to know how to use the distributive property to solve an equation, just follow the steps below.
Steps
Method 1 of 4: How to Use Distributive Property: Elementary Case
Step 1. Multiply the term outside the parentheses with the terms inside the parentheses
In doing this, you are essentially distributing the term that is outside the brackets to those that are inside. Multiply the outer term by the first of the inner terms and then by the second. If there are more than two, continue applying the property by multiplying by the remaining terms. Here's how to do it:
- Ex: 2 (x - 3) = 10
- 2 (x) - (2) (3) = 10
- 2x - 6 = 10
Step 2. Add the like terms
Before solving the equation you will need to add up the similar terms. Add up all numeric terms and all terms that contain "x". Move all numeric terms to the right of the equal and all terms with "x" to the left.
- 2x - 6 (+6) = 10 (+6)
- 2x = 16
Step 3. Solve the equation
Find the value of "x" by dividing both terms of the equation by 2.
- 2x = 16
- 2x / 2 = 16/2
- x = 8
Method 2 of 4: How to Use Distributive Property: Most Advanced Case
Step 1. Multiply the term outside the parentheses with the terms inside the parentheses
This step is the same as we did in the base case, but in this case you will be using the distributive property more than once in the same equation.
- Ex: 4 (x + 5) = 8 + 6 (2x - 2)
- 4 (x) + 4 (5) = 8 + 6 (2x) - 6 (2)
- 4x + 20 = 8 + 12x -12
Step 2. Add the like terms
Add up all similar terms and move them so that all terms containing x are to the left of the equal and all numeric terms are to the right.
- 4x + 20 = 8 + 12x -12
- 4x + 20 = 12x - 4
- 4x -12x = -4 - 20
- -8x = -24
Step 3. Solve the equation
Find the value of "x" by dividing both terms of the equation by -8.
- -8x / -8 = -24 / -8
- x = 3
Method 3 of 4: How to Apply Distributive Property with a Negative Coefficient
Step 1. Multiply the term outside the parentheses with the terms inside
If it has a negative sign, simply distribute the sign as well. If you are multiplying a negative number by a positive one, the result will be negative; if you are multiplying a negative number by another negative number, the result will be positive.
- Ex: -4 (9 - 3x) = 48
- -4 (9) - [-4 (3x)] = 48
- -36 - (- 12x) = 48
- -36 + 12x = 48
Step 2. Add the like terms
Move all terms with "x" to the left of the equal and all numeric terms to the right.
- -36 + 12x = 48
- 12x = 48 - [- (36)]
- 12x = 84
Step 3. Solve the equation
Find the value of "x" by dividing both terms of the equation by 12.
- 12x / 12 = 84/12
- x = 7
Method 4 of 4: How to Simplify Denominators in an Equation
Step 1. Find the least common multiple (lcm) of the denominators of the fractions in the equation
To find the lcm, you need to find the smallest number that is a multiple of all the denominators of the fractions in the equation. The denominators are 3 and 6; 6 is the smallest number which is a multiple of both 3 and 6.
- x - 3 = x / 3 + 1/6
- mcm = 6
Step 2. Multiply the terms of the equation by the lcm
Now put all the terms on the left of the equation in brackets and do the same for those on the right, and put the lcm outside the brackets. Then multiply, applying the distributive property if necessary. Multiplying both terms of the brackets by the same number turns the equation into an equivalent, that is, into another equation that has the same result, but has numbers that are easier to calculate with after you have simplified the fractions.
- 6 (x - 3) = 6 (x / 3 + 1/6)
- 6 (x) - 6 (3) = 6 (x / 3) + 6 (1/6)
- 6x - 18 = 2x + 1
Step 3. Add the like terms
Move all terms with "x" to the left of the equal and all numeric terms to the right.
- 6x - 2x = 1 - (-18)
- 4x = 19
Step 4. Solve the equation
Find the value of "x" by dividing both terms by 4.
- 4x / 4 = 19/4
- x = 19/4 or (16 + 3) / 4
