Solving an equation using the distributive property is related to the need to open the parentheses as the first step to then be able to simplify similar members. When an equation contains one or more pairs of parentheses, we must start by opening them all and then proceed to the next phase. 

Below, we provide you with some examples where this method is applied.

2(X+3)=8 2\left(X+3\right)=8

In this equation, we can clearly see some parentheses. To start, we must open them (that is, apply the distributive property) and then we can proceed with the following phases of the exercise.

2X+6=8 2X+6=8

2X=2 2X=2

X=1 X=1

The result of the equation is 1 1 .

Solving equations using the distributive property


Suggested Topics to Practice in Advance

  1. Solving Equations by Adding or Subtracting the Same Number from Both Sides
  2. Solving Equations by Multiplying or Dividing Both Sides by the Same Number
  3. Solving Equations by Simplifying Like Terms

Practice Solving Quadratic Equations using Factoring

Examples with solutions for Solving Quadratic Equations using Factoring

Exercise #1

Determine the value of x x :

2(x+4)+8=0 2(x+4)+8=0

Video Solution

Step-by-Step Solution

Let's first expand the parentheses using the formula:

a(x+b)=ax+ab a(x+b)=ax+ab

(2×x)+(2×4)+8=0 (2\times x)+(2\times4)+8=0

2x+8+8=0 2x+8+8=0

Next, we will substitute in our terms accordingly:

2x+16=0 2x+16=0

Then, we will move the 16 to the left-hand side, keeping the appropriate sign:

2x=16 2x=-16

Finally, we divide both sides by 2:

2x2=162 \frac{2x}{2}=-\frac{16}{2}

x=8 x=-8

Answer

x=8 x=-8

Exercise #2

3(a+1)3=0 3(a+1)-3=0

Video Solution

Step-by-Step Solution

Let's proceed to solve the linear equation 3(a+1)3=0 3(a+1) - 3 = 0 :

Step 1: Distribute the 3 in the expression 3(a+1) 3(a+1) .

We get:
3a+313=0 3 \cdot a + 3 \cdot 1 - 3 = 0

This simplifies to:
3a+33=0 3a + 3 - 3 = 0

Step 2: Simplify the expression by combining like terms.

We simplify this to:
3a+0=0 3a + 0 = 0 or simply 3a=0 3a = 0

Step 3: Isolate a a by dividing both sides by 3.

3a3=03\frac{3a}{3} = \frac{0}{3}

Thus,
a=0 a = 0

Therefore, the solution to the problem is a=0 a = 0 .

The correct choice is the option corresponding to a=0 a = 0 .

Answer

a=0 a=0

Exercise #3

Solve for x:

7(2x+5)=77 7(-2x+5)=77

Video Solution

Step-by-Step Solution

To open parentheses we will use the formula:

a(x+b)=ax+ab a(x+b)=ax+ab

(7×2x)+(7×5)=77 (7\times-2x)+(7\times5)=77

We multiply accordingly

14x+35=77 -14x+35=77

We will move the 35 to the right section and change the sign accordingly:

14x=7735 -14x=77-35

We solve the subtraction exercise on the right side and we will obtain:

14x=42 -14x=42

We divide both sections by -14

14x14=4214 \frac{-14x}{-14}=\frac{42}{-14}

x=3 x=-3

Answer

-3

Exercise #4

Solve for y:

2(4+y)y=0 -2(-4+y)-y=0

Video Solution

Step-by-Step Solution

To solve the equation 2(4+y)y=0-2(-4 + y) - y = 0, we will follow these steps:

  • Step 1: Distribute 2 -2 inside the parenthesis.
  • Step 2: Simplify and combine like terms.
  • Step 3: Solve the equation for yy.

Let's proceed with the solution:

Step 1: Distribute 2-2 in the expression 2(4+y)-2(-4 + y). This will transform the expression as follows:

2(4+y)=2×4+(2)×y=82y-2(-4 + y) = -2 \times -4 + (-2) \times y = 8 - 2y.

After distributing, the equation becomes:

82yy=08 - 2y - y = 0.

Step 2: Combine like terms. Notice that 2yy-2y - y is equivalent to 3y-3y:

83y=08 - 3y = 0.

Step 3: Solve for yy. First, isolate the term with yy by subtracting 8 from both sides:

3y=8-3y = -8.

Next, divide both sides by 3-3 to find yy:

y=83=83y = \frac{-8}{-3} = \frac{8}{3}.

Thus, the solution for yy is 83\frac{8}{3}, which can be written as a mixed number:

y=223y = 2\frac{2}{3}.

Therefore, the solution to the problem is y=223y = 2\frac{2}{3}.

Answer

y=223 y=2\frac{2}{3}

Exercise #5

5(3b1)=0 5-(3b-1)=0

Video Solution

Step-by-Step Solution

To solve the given linear equation 5(3b1)=0 5 - (3b - 1) = 0 , follow these steps:

  • Step 1: Simplify the equation.
    Start by distributing the negative sign through the parentheses:
    53b+1=0 5 - 3b + 1 = 0
  • Step 2: Combine like terms.
    Combine the constant terms on the left side:
    63b=0 6 - 3b = 0
  • Step 3: Isolate the variable b b .
    Subtract 6 from both sides of the equation to isolate the term with b b :
    3b=6-3b = -6
  • Step 4: Solve for b b .
    Divide both sides by -3 to solve for b b :
    b=63=2 b = \frac{-6}{-3} = 2

Therefore, the solution to the equation is b=2 b = 2 .

Answer

b=2 b=2

Exercise #6

8a+2(3a7)=0 8a+2(3a-7)=0

Video Solution

Step-by-Step Solution

To solve the linear equation 8a+2(3a7)=0 8a + 2(3a - 7) = 0 , we'll proceed with the following steps:

Step 1: Apply the Distributive Property.
The equation given is 8a+2(3a7)=0 8a + 2(3a - 7) = 0 .
First, distribute the 2 across the terms inside the parenthesis:
2(3a7)=2×3a+2×(7)=6a14 2(3a - 7) = 2 \times 3a + 2 \times (-7) = 6a - 14 .
By substituting this back into the equation, we have:
8a+6a14=0 8a + 6a - 14 = 0 .

Step 2: Combine Like Terms.
Now, combine the terms containing a a :
8a+6a=14a 8a + 6a = 14a .
The equation now becomes:
14a14=0 14a - 14 = 0 .

Step 3: Isolate the Variable.
Add 14 to both sides of the equation to isolate terms with a a :
14a14+14=0+14 14a - 14 + 14 = 0 + 14 , which simplifies to:
14a=14 14a = 14 .
Next, divide both sides by 14 to solve for a a :
a=1414=1 a = \frac{14}{14} = 1 .

Therefore, the solution to the equation is a=1 a = 1 .

Answer

a=1 a=1

Exercise #7

Solve for x:

2(4x)=8 2(4-x)=8

Video Solution

Step-by-Step Solution

To solve this equation, follow these steps:

  • Step 1: Apply the distributive property to the equation:

    2(4x)=2×42×x=82x 2(4-x) = 2 \times 4 - 2 \times x = 8 - 2x

  • Step 2: Simplify the equation:

    The equation now becomes: 82x=88 - 2x = 8

  • Step 3: Isolate the variable xx by simplifying the equation:

    First, subtract 8 from both sides:
    82x8=88 8 - 2x - 8 = 8 - 8
    This simplifies to:
    2x=0-2x = 0

  • Step 4: Solve for xx by dividing both sides by -2:

    x=02=0 x = \frac{0}{-2} = 0

Therefore, the solution to the equation is x=0x = 0.

Answer

0

Exercise #8

3x+5(x+4)=0 3x+5(x+4)=0

Video Solution

Step-by-Step Solution

To solve this problem, we'll follow these steps:

  • Step 1: Apply the distributive property to the equation.
  • Step 2: Combine like terms.
  • Step 3: Solve for x x .

Now, let's work through each step:

Step 1: Distribute the number 5 across the expression inside the parentheses:
3x+5(x+4)=0 3x + 5(x + 4) = 0 becomes 3x+5x+20=0 3x + 5x + 20 = 0 .

Step 2: Combine the like terms:
Combine 3x 3x and 5x 5x to get 8x 8x .
Thus, the equation becomes 8x+20=0 8x + 20 = 0 .

Step 3: Solve for x x :
Subtract 20 from both sides: 8x=20 8x = -20 .
Finally, divide both sides by 8: x=208 x = \frac{-20}{8} .

Simplify the fraction: x=2.5 x = -2.5 .

Therefore, the solution to the equation is x=2.5 x = -2.5 .

Answer

x=2.5 x=-2.5

Exercise #9

Solve x:

5(x+3)=0 5(x+3)=0

Video Solution

Step-by-Step Solution

We open the parentheses according to the formula:

a(x+b)=ax+ab a(x+b)=ax+ab

5×x+5×3=0 5\times x+5\times3=0

5x+15=0 5x+15=0

We will move the 15 to the right section and keep the corresponding sign:

5x=15 5x=-15

Divide both sections by 5

5x5=155 \frac{5x}{5}=\frac{-15}{5}

x=3 x=-3

Answer

3 -3

Exercise #10

Solve for X:

12(x+3)8x=3 \frac{1}{2}(x+3)-8x=3

Video Solution

Step-by-Step Solution

Let's solve the given equation 12(x+3)8x=3 \frac{1}{2}(x+3) - 8x = 3 .

First, distribute the 12\frac{1}{2} inside the parentheses:
12(x+3)=12x+12×3=12x+32 \frac{1}{2}(x+3) = \frac{1}{2}x + \frac{1}{2} \times 3 = \frac{1}{2}x + \frac{3}{2} .

Substitute back into the original equation:
12x+328x=3 \frac{1}{2}x + \frac{3}{2} - 8x = 3 .

To eliminate the fraction, multiply every term by 2 to simplify:
2×(12x+32)2×8x=2×3 2 \times \left(\frac{1}{2}x + \frac{3}{2}\right) - 2 \times 8x = 2 \times 3 .

After clearing the fraction, the equation becomes:
x+316x=6 x + 3 - 16x = 6 .

Combine like terms involving x x :
x16x+3=6 x - 16x + 3 = 6 simplifies to 15x+3=6 -15x + 3 = 6 .

Isolate x x by subtracting 3 from both sides:
15x=63 -15x = 6 - 3 .
This simplifies to 15x=3 -15x = 3 .

Finally, divide both sides by 15-15 to solve for x x :
x=315 x = \frac{3}{-15} ,
which simplifies to x=15 x = -\frac{1}{5} .

Therefore, the solution to the equation 12(x+3)8x=3 \frac{1}{2}(x+3) - 8x = 3 is x=15 x = -\frac{1}{5} .

Answer

15 -\frac{1}{5}

Exercise #11

Solve the following exercise:

3(4a+8)=27a -3(4a+8)=27a

a=? a=\text{?}

Video Solution

Step-by-Step Solution

To open the parentheses on the left side, we'll use the formula:

a(b+c)=abac -a\left(b+c\right)=-ab-ac

12a24=27a -12a-24=27a

We'll arrange the equation so that the terms with 'a' are on the right side, and maintain the plus and minus signs during the transfer:

24=27a+12a -24=27a+12a

Let's group the terms on the right side:

24=39a -24=39a

Let's divide both sides by 39:

2439=39a39 -\frac{24}{39}=\frac{39a}{39}

2439=a -\frac{24}{39}=a

Note that we can reduce the fraction since both numerator and denominator are divisible by 3:

813=a -\frac{8}{13}=a

Answer

813 -\frac{8}{13}

Exercise #12

Solve for X:

12(x+3)=0 \frac{1}{2}(x+3)=0

Video Solution

Step-by-Step Solution

To solve for x x in the equation 12(x+3)=0 \frac{1}{2}(x+3) = 0 , we will take the following steps:

  • Step 1: Eliminate the fraction by multiplying both sides of the equation by 2.

  • Step 2: Simplify the resulting equation to isolate the variable x x .

Let's carry out these steps:
Step 1: Multiply both sides by 2:
2×12(x+3)=2×0\Rightarrow 2 \times \frac{1}{2}(x + 3) = 2 \times 0
x+3=0\Rightarrow x + 3 = 0

Step 2: Solve the equation x+3=0 x + 3 = 0 for x x :
Subtract 3 from both sides:
x=03\Rightarrow x = 0 - 3
x=3\Rightarrow x = -3

Therefore, the solution to the equation is x=3 x = -3 .

Answer

3-

Exercise #13

6c+7+4c=3(c1) 6c+7+4c=3(c-1)

c=? c=\text{?}

Video Solution

Step-by-Step Solution

To solve the equation 6c+7+4c=3(c1) 6c + 7 + 4c = 3(c - 1) , follow these steps:

  • Step 1: Combine like terms on the left side of the equation.
    The like terms are 6c6c and 4c4c. Combining these gives 10c+7=3(c1)10c + 7 = 3(c - 1).
  • Step 2: Apply the distributive property on the right side of the equation.
    The term 3(c1)3(c - 1) expands to 3c33c - 3. Therefore, the equation becomes 10c+7=3c310c + 7 = 3c - 3.
  • Step 3: Move all terms involving cc to one side and constants to the other.
    Subtract 3c3c from both sides: 10c3c+7=310c - 3c + 7 = -3 which simplifies to 7c+7=37c + 7 = -3.
  • Step 4: Isolate the term with cc by subtracting 7 from both sides of the equation.
    This gives 7c=377c = -3 - 7 or 7c=107c = -10.
  • Step 5: Solve for cc.
    Divide both sides by 7: c=107=107c = \frac{-10}{7} = -\frac{10}{7}. This can be converted to a mixed number, giving 137-1\frac{3}{7}.

Therefore, the solution to the equation is c=137 c = -1\frac{3}{7} . This corresponds to choice 2 in the provided answer choices.

Answer

137 -1\frac{3}{7}

Exercise #14

7y+10y+5=2(y+3) 7y+10y+5=2(y+3)

y=? y=\text{?}

Video Solution

Step-by-Step Solution

To solve the equation 7y+10y+5=2(y+3) 7y + 10y + 5 = 2(y + 3) , let's proceed as follows:

  • Step 1: Simplify the left side by combining like terms. The expression 7y+10y 7y + 10y combines to 17y 17y , so we have 17y+5=2(y+3) 17y + 5 = 2(y + 3) .

  • Step 2: Expand the right side. Distribute the 2 across the parenthesis: 2(y+3) 2(y + 3) becomes 2y+6 2y + 6 . The equation now reads 17y+5=2y+6 17y + 5 = 2y + 6 .

  • Step 3: Isolate terms involving y y on one side. Subtract 2y 2y from both sides: 17y2y+5=6 17y - 2y + 5 = 6 , which simplifies to 15y+5=6 15y + 5 = 6 .

  • Step 4: Isolate 15y 15y by subtracting 5 from both sides: 15y=65 15y = 6 - 5 , which simplifies to 15y=1 15y = 1 .

  • Step 5: Solve for y y by dividing both sides by 15: y=115 y = \frac{1}{15} .

Therefore, the solution to the problem is y=115 \mathbf{y = \frac{1}{15}} .

Answer

115 \frac{1}{15}

Exercise #15

Solve for X:

8(2x)=12+x -8(2-x)=\frac{1}{2}+x

Video Solution

Step-by-Step Solution

To solve the equation 8(2x)=12+x -8(2-x) = \frac{1}{2} + x , we will follow these detailed steps:

  • Step 1: Distribute the 8-8 to both terms inside the parentheses:
    8(2x)=8×2+(8)×(x)=16+8x -8(2-x) = -8 \times 2 + (-8) \times (-x) = -16 + 8x .
  • Step 2: Rewrite the equation from our distribution:
    16+8x=12+x -16 + 8x = \frac{1}{2} + x .
  • Step 3: Move all x x -terms to one side and constants to the other. Subtract x x from both sides:
    8xx=12+16 8x - x = \frac{1}{2} + 16 .
  • Step 4: Simplify both sides:
    7x=12+16 7x = \frac{1}{2} + 16 .
  • Step 5: Combine like terms on the right side:
    Convert 16 16 to an equivalent fraction of 322\frac{32}{2} so we can sum with 12\frac{1}{2}:
    7x=12+322=332 7x = \frac{1}{2} + \frac{32}{2} = \frac{33}{2} .
  • Step 6: Solve for x x by dividing by 7:
    x=332×17=3314 x = \frac{33}{2} \times \frac{1}{7} = \frac{33}{14} .

Therefore, the solution to the equation is x=3314 x = \frac{33}{14} .

The choice : 3314 \frac{33}{14}

matches our solution.

Answer

3314 \frac{33}{14}