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• To step
• Method 1 of 4: Using the basic distributive property
• Tips

The distributive property is a rule of mathematics for simplifying an equation with parentheses. You probably learned early on to do the operations in parentheses first, but algebraic expressions don't always do that. The distributive property allows you to multiply the term outside parentheses by the terms inside it. You have to make sure you do it the right way, otherwise you can lose information and the comparison will no longer be correct. You can also use the distributive property to simplify equations with fractions.

To step

Method 1 of 4: Using the basic distributive property

1. Multiply the term outside parentheses by each term in parentheses. To do this, essentially divide the outer term among the inner terms. Multiply the term outside parentheses by the first term in parentheses. Then you multiply it by the second term. If there are more than two terms, keep distributing the term outside of the parentheses, over all of the terms inside the parenthesis. Just leave the operators (plus or minus) inside the brackets.
   • ${ displaystyle 2 (x-3) = 10}$Combine like terms. Before you can solve the equation, you have to combine like terms. Combine all numeric terms. In addition, you combine all variable terms separately. To simplify the equation, order the terms so that the variables are on one side of the equal sign and the constants (numbers only) on the other.
     • ${ displaystyle 2x-6 = 10}$Solve the equation. Loose ${ displaystyle x}$Distribute a negative number along with the minus sign. If you are going to multiply a term or terms in parentheses by a negative number, make sure to apply the minus sign to each term within the parentheses.
       • Remember the basic rules for multiplying with negative numbers:
         • Minus x Minus = Plus.
         • Minus x Plus = Min.
       • Consider the following example:
         • ${ displaystyle -4 (9-3x) = 48}$Combine like terms. After you have completed the distribution, you then need to simplify the equation by moving all variable terms to one side of the equal sign, and all numbers without variables to the other. You do this by means of a combination of addition or subtraction.
           • ${ displaystyle -36 + 12x = 48}$Share to get the final solution. Solve the equation by dividing both sides of the equation by the coefficient of the variable. This should result in a single variable on one side of the equation, with the result on the other.
             • ${ displaystyle 12x = 84}$Treat subtraction as addition (from -1). When you see a minus sign in an algebra problem, especially if it is before a parenthesis, it essentially says + (-1). This helps distribute the minus sign correctly across all parenthetical terms. Then solve the problem as before.
               • For example, consider the problem, ${ displaystyle 4x- (x + 2) = 4}$Check for fractional coefficients or constants. Sometimes you may have to solve a problem with fractions as coefficients or constants. You can leave them as they are and apply the basic rules of algebra to solve the problem. However, by taking advantage of the distributive property, you can often simplify the solution by converting fractions to integers.
                 • Consider the following example ${ displaystyle x-3 = { frac {x} {3}} + { frac {1} {6}}}$Find the least common multiple (LCM) for all denominators. You can ignore all integers at this step. Look only at the fractions and determine the lcm for all denominators. Find the LC by looking for the smallest number that is a multiple of the denominators of both fractions in the equation. In this example, the denominators are 3 and 6, so 6 is the LCM.
                 • Multiply all terms of the equation by the LCM. Remember, you can apply any operation to a math equation as long as you do it on both sides. By multiplying each term of the equation by the LCM, the terms will cancel each other out and become "" integers. Place your parentheses around the entire left and right sides of the equation, then do the distribution:
                   • ${ displaystyle x-3 = { frac {x} {3}} + { frac {1} {6}}}$Combine like terms. Combine all terms so that all variables are on one side of the equation and all constants on the other. Use the basic addition and subtraction operations to move terms from one side to the other of the equation.
                     • ${ displaystyle 6x-18 = 2x + 1}$Solve the equation. Find the final solution by dividing both sides of the equation by the coefficient of the variable. This leaves x on one side of the equation and the numerical solution on the other.
                       • ${ displaystyle 4x = 19}$Interpret a fraction with an equation as a distributed division. Sometimes you see a problem with multiple terms in the numerator of a fraction, above a common denominator. You have to treat this as a distributive problem and apply the denominator to every term of the numerator. You can rewrite the fraction to show the distribution. As follows:
                         • ${ displaystyle { frac {4x + 8} {2}} = 4}$Simplify each numerator as a separate fraction. After distributing the divisor over each term, you can then simplify each term individually.
                           • ${ displaystyle { frac {4x} {2}} + { frac {8} {2}} = 4}$Isolate the variable. Continue to solve the problem by isolating the variable on one side of the equation and moving the constant terms to the other. Do this through a combination of addition and subtraction, as needed.
                             • ${ displaystyle 2x + 4 = 4}$Divide by the coefficient to solve the problem. In the last step, you divide by the coefficient of the variable. This gives the final solution, with the single variable on one side of the equation and the numerical solution on the other.
                               • ${ displaystyle 2x = 0}$Avoid the common mistake of sharing just one term. It's tempting (but incorrect) to divide the first term of the numerator by the denominator and work out the fraction. An error like this would look like this for the above problem:
                                 • ${ displaystyle { frac {4x + 8} {2}} = 4}$Check the correctness of your solution. You can always check your work by inserting your solution into the original problem. If you want to simplify, you have to come up with a true statement. If you simplify and get an incorrect statement as the answer, then your solution is incorrect. In this example, you test the two solutions for x = 0 and x = -2 to see which one is correct.
                                   • Start with solution x = 0:
                                     • ${ displaystyle { frac {4x + 8} {2}} = 4}$..... (original problem)
                                     • ${ displaystyle { frac {4 (0) +8} {2}} = 4}$..... (substitute 0 for x)
                                     • ${ displaystyle { frac {0 + 8} {2}} = 4}$
                                     • ${ displaystyle { frac {8} {2}} = 4}$
                                     • ${ displaystyle 4 = 4}$..... (True. This is the right solution.)
                                   • Try the "incorrect solution for x = -2:
                                     • ${ displaystyle { frac {4x + 8} {2}} = 4}$..... (original problem)
                                     • ${ displaystyle { frac {4 (-2) +8} {2}} = 4}$..... (enter -2 for x)
                                     • ${ displaystyle { frac {-8 + 8} {2}} = 4}$
                                     • ${ displaystyle { frac {0} {2}} = 4}$
                                     • ${ displaystyle 0 = 4}$..... (False statement. Therefore x = -2 is false.)

Tips

• You can also use the distributive property to simplify some multiplications. You can divide numbers into tens with a remainder to make mental arithmetic easier. For example, you can rewrite 8 x 16 as 8 (10 + 6). This is just 80 + 48 = 128. Another example, 7 x 24 = 7 (20 + 4) = 7 (20) + 7 (4) = 140 + 28 = 168. Practice these by heart and mental arithmetic will a lot easier.