# 4 Ways to Use Distributive Property to Solve an Equation

The distributive property represents a mathematical rule that helps to simplify equations with parentheses. You learned a long time ago that you should start with operations inside parentheses, but that's not always possible in algebraic expressions. The distributive property allows you to multiply the term outside the parentheses by those inside it. You have to do it well so you don't lose information and solve the equation correctly. It is also possible to use the distributive property to simplify equations involving fractions.

## Steps

### Method 1 of 4: Using the Basic Distributive Property

#### Step 1. Multiply the term outside the parentheses by each term inside it

Essentially, for that, you're distributing the external term over the internal ones. First, multiply the term outside the parentheses by the first term inside them. Then multiply it by the second term. If there are more than two terms, continue multiplying until there are none left. Keep the signs (positive or negative) of each element in parentheses.

• 2(x−3)=10{displaystyle 2(x-3)=10}

• 2(x)−(2)(3)=10{displaystyle 2(x)-(2)(3)=10}
• 2x−6=10{displaystyle 2x-6=10}

#### Step 2. Combine similar terms

Before solving the equation, you will have to combine similar terms. Match all the numeric terms to each other. Separately, do this with any unknowns present. To simplify the equation, arrange the terms so that variables are on one side of the equal sign and constants (only numbers) are on the other.

• 2x−6=10{displaystyle 2x-6=10}

- (problema original)

• 2x−6(+6)=10(+6){displaystyle 2x-6(+6)=10(+6)}
• - (some 6 a ambos os lados)

• 2x=16{displaystyle 2x=16}
• - (variável na esquerda e constante na direita)

#### Step 3. Solve the equation

Divide by x{displaystyle x}

, dividindo ambos os lados da equação pelo coeficiente à frente da variável.

• 2x=16{displaystyle 2x=16}
• - (problema original)

• 2x/2=16/2{displaystyle 2x/2=16/2}
• - (divida ambos os lados por 2)

• x=8{displaystyle x=8}
• - (solução)

### Método 2 de 4: Distribuindo coeficientes negativos

#### Step 1. Distribute a negative number along with its negative sign

If you have a negative number by multiplying one or more terms inside the parentheses, distribute the negative sign over the inner numbers as well.

• Remember the basic rules of multiplication with negatives:

• Consider the following example:

• −4(9−3x)=48{displaystyle -4(9-3x)=48}

- (problema original)

• −4(9)−(−4)(3x)=48{displaystyle -4(9)-(-4)(3x)=48}
• - (distribua (-4) para cada termo)

• −36−(−12x)=48{displaystyle -36-(-12x)=48}
• - (simplifique a multiplicação)

• −36+12x=48{displaystyle -36+12x=48}
• - (observe que 'menos -12' se torna +12)

#### Step 2. Combine similar terms

After you finish the distribution, you'll have to simplify the equation by passing all variable terms to one side of the equal sign and all non-variable terms to the other. Do it with a combination of addition or subtraction.

• −36+12x=48{displaystyle -36+12x=48}

- (problema original)

• −36(+36)+12x=48+36{displaystyle -36(+36)+12x=48+36}

• 12x=84{displaystyle 12x=84}
• - (simplifique a soma para isolar a incógnita)

#### Step 3. Divide to find the final solution

Solve the equation by dividing both sides by the coefficient of the variable. This will result in a single unknown on one side of the equation with the result on the other.

• 12x=84{displaystyle 12x=84}

- (problema original)

• 12x/12=84/12{displaystyle 12x/12=84/12}
• - (divida ambos os lados por 12)

• x=7{displaystyle x=7}
• - (solução)

#### Step 4. Treat subtraction as the sum of (-1)

When faced with a negative sign in an algebraic problem, especially if it comes before the parentheses, imagine that it reads + (-1). This will help you correctly distribute negative values ​​to all terms within the parentheses. Then solve the problem normally.

• For example, consider the problem, 4x−(x+2)=4{displaystyle 4x-(x+2)=4}

. Para distribuir os negativos corretamente, reescreva o problema para que se leia:

• 4x+(−1)(x+2)=4{displaystyle 4x+(-1)(x+2)=4}
• A seguir, distribua o (-1) aos termos internos dos parênteses, como se segue:

• 4x+(−1)(x+2)=4{displaystyle 4x+(-1)(x+2)=4}

• 4x−x−2=4{displaystyle 4x-x-2=4}
• - (multiplique (-1) por x e por 2)

• 3x−2=4{displaystyle 3x-2=4}
• - (combine os termos)

• 3x−2+2=4+2{displaystyle 3x-2+2=4+2}
• - (some 2 a ambos os lados)

• 3x=6{displaystyle 3x=6}
• - (simplifique os termos)

• 3x/3=6/3{displaystyle 3x/3=6/3}
• - (divida ambos os lados por 3)

• x=2{displaystyle x=2}
• - (solução)

### Método 3 de 4: Usando a propriedade distributiva para simplificar as frações

#### Step 1. Identify any fractional coefficients or constants

Sometimes you might have a problem that contains fractions such as coefficients or constants. You can leave them as they are and apply the basic rules of algebra to solve the problem. However, using the distributive property can also simplify the solution by turning fractions into whole numbers.

• Consider the example x−3=x3+16{displaystyle x-3={frac {x}{3}}+{frac {1}{6}}}

. As frações nesse problema são x3{displaystyle {frac {x}{3}}}

e 16{displaystyle {frac {1}{6}}}

#### Step 2. Find the least common multiple (MMC) for all denominators

In this step, you can ignore all whole numbers. Just look for fractions and calculate the MMC of all denominators. To calculate the MMC, you must find the smallest number equally divisible by all the denominators of the fractions. In this example, the denominators are 3 and 6, so the MMC will be 6.

#### Step 3. Multiply all terms in the equation by the MMC

Remember that you can perform any operation you want on an algebraic equation as long as you do it on both sides. Multiply all the terms in the equation by the MMC, and the fractions will cancel out to "become" whole numbers. Put parentheses around the values ​​on the right and left side and then distribute:

• x−3=x3+16{displaystyle x-3={frac {x}{3}}+{frac {1}{6}}}

- (equação original)

• (x−3)=(x3+16){displaystyle (x-3)=({frac {x}{3}}+{frac {1}{6}})}
• - (coloque os parênteses)

• 6(x−3)=6(x3+16){displaystyle 6(x-3)=6({frac {x}{3}}+{frac {1}{6}})}
• - (multiplique ambos os lados pelo MMC)

• 6x−6(3)=6(x3)+6(16){displaystyle 6x-6(3)=6({frac {x}{3}})+6({frac {1}{6}})}
• - (distribua a multiplicação)

• 6x−18=2x+1{displaystyle 6x-18=2x+1}
• - (simplifique a multiplicação)

#### Step 4. Combine similar terms

Combine all the terms so that the variables are on one side of the equation and all the constants are on the other. Use basic addition and subtraction operations to move terms back and forth.

• 6x−18=2x+1{displaystyle 6x-18=2x+1}

• 6x−2x−18=2x−2x+1{displaystyle 6x-2x-18=2x-2x+1}
• - (subtraia 2x de ambos os lados)

• 4x−18=1{displaystyle 4x-18=1}
• - (simplifique a subtração)

• 4x−18+18=1+18{displaystyle 4x-18+18=1+18}
• - (some 18 a ambos os lados)

• 4x=19{displaystyle 4x=19}
• - (simplifique a soma)

#### Step 5. Solve the equation

Find the final solution by dividing both sides of the equation by the coefficient of the variable. This should leave a single x term on one side of the equation, with the numerical solution on the other.

• 4x=19{displaystyle 4x=19}

• 4x/4=19/4{displaystyle 4x/4=19/4}
• - (divida ambos os lados por 4)

• x=194{displaystyle x={frac {19}{4}}}
• ou 434{displaystyle 4{frac {3}{4}}}

### Método 4 de 4: Distribuindo uma fração longa

#### Step 1. Interpret a long fraction in the form of distributed division

Sometimes you may see a problem that contains multiple terms in the numerator of a fraction over a single denominator. You should treat it as if it were a distributive problem, applying the denominator to each term in the numerator. Rewrite the fraction to display this distribution as follows:

• 4x+82=4{displaystyle {frac {4x+8}{2}}=4}

- (problema original)

• 4x2+82=4{displaystyle {frac {4x}{2}}+{frac {8}{2}}=4}

#### Step 2. Simplify each numerator as a separate fraction

After distributing the denominator to each of the terms, you can simplify them individually.

• 4x2+82=4{displaystyle {frac {4x}{2}}+{frac {8}{2}}=4}

• 2x+4=4{displaystyle 2x+4=4}
• - (simplifique as frações)

#### Step 3. Isolate the variable

Continue to solve the problem by isolating the variable on one side of the equation and passing the constant terms to the other side. Do this with a combination of additions and subtractions as needed.

• 2x+4=4{displaystyle 2x+4=4}

• 2x+4−4=4−4{displaystyle 2x+4-4=4-4}
• - (subtraia 4 de ambos os lados)

• 2x=0{displaystyle 2x=0}

#### Step 4. Divide by the coefficient to solve the problem

In the final step, divide by the coefficient of the variable. This will bring up the final solution, with the single variable on one side of the equation and the numerical solution on the other.

• 2x=0{displaystyle 2x=0}

• 2x2=02{displaystyle {frac {2x}{2}}={frac {0}{2}}}
• - (divida ambos os lados por 2)

• x=0{displaystyle x=0}
• - (solução)

#### Step 5. Avoid the common mistake of dividing by just one term

It is tempting (but incorrect) to divide the first numerator by the denominator and cancel the fraction. An error like this, in the problem above, would look like this:

• 4x+82=4{displaystyle {frac {4x+8}{2}}=4}

- (problema original)

• 2x+8=4{displaystyle 2x+8=4}
• - (divida apenas 4x por 2 em vez de pelo numerador completo)

• 2x+8−8=4−8{displaystyle 2x+8-8=4-8}
• 2x=−4{displaystyle 2x=-4}
• x=−2{displaystyle x=-2}
• - (solução incorreta)

#### Step 6. Check if the solution is correct

You can always check the calculations made by inserting the solution into the original problem. By simplifying it, you must arrive at a true statement. If the simplification results in a false statement, this indicates that the solution was incorrect. In this example, test both solutions (x=0 and x=-2) to see which one is correct.

• 4x+82=4{displaystyle {frac {4x+8}{2}}=4}

- (problema original)

• 4(0)+82=4{displaystyle {frac {4(0)+8}{2}}=4}
• - (coloque 0 no lugar de x)

• 0+82=4{displaystyle {frac {0+8}{2}}=4}
• 82=4{displaystyle {frac {8}{2}}=4}
• 4=4{displaystyle 4=4}
• 4x+82=4{displaystyle {frac {4x+8}{2}}=4}

• 4(−2)+82=4{displaystyle {frac {4(-2)+8}{2}}=4}
• −8+82=4{displaystyle {frac {-8+8}{2}}=4}
• 02=4{displaystyle {frac {0}{2}}=4}
• 0=4{displaystyle 0=4}