Solve -3(4a+8)=27a: Linear Equation with Distributive Property

Linear Equations with Distribution and Fractions

Solve the following exercise:

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

a=? a=\text{?}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve
00:04 Open parentheses properly, multiply by each factor
00:07 We want to isolate the unknown A
00:14 Arrange the equation so that one side has only A
00:26 Collect terms
00:32 Isolate the unknown A
00:42 Factor 24 into 3 and 8
00:47 Factor 39 into 3 and 13
00:53 Simplify what we can
01:01 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution
1

Understand the problem

Solve the following exercise:

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

a=? a=\text{?}

2

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

3

Final Answer

813 -\frac{8}{13}

Key Points to Remember

Essential concepts to master this topic
  • Distribution Rule: Multiply -3 by each term: -3(4a+8) = -12a-24
  • Technique: Move all a terms to one side: -24 = 27a + 12a = 39a
  • Check: Substitute a=813 a = -\frac{8}{13} : both sides equal 7213 -\frac{72}{13}

Common Mistakes

Avoid these frequent errors
  • Forgetting to distribute negative sign to all terms
    Don't write -3(4a+8) = -12a+24! This changes the sign of 24 incorrectly and gives the wrong equation. The negative multiplies both terms, so always write -12a-24.

Practice Quiz

Test your knowledge with interactive questions

\( \frac{-y}{5}=-25 \)

FAQ

Everything you need to know about this question

Why do I get a fraction instead of a whole number?

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Many linear equations have fractional solutions! This is completely normal. The key is to simplify the fraction to its lowest terms, like reducing 2439 -\frac{24}{39} to 813 -\frac{8}{13} .

How do I know when to distribute the negative sign?

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When you see a negative number in front of parentheses, it multiplies every term inside. Think of -3(4a+8) as (-3)×(4a) + (-3)×(8) = -12a + (-24) = -12a - 24.

Should I move the a terms to the left or right side?

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Either works! Choose the side that keeps your coefficient positive when possible. In this problem, moving a terms to the right gives 39a, which is easier than -39a.

How can I check if my fraction answer is correct?

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Substitute a=813 a = -\frac{8}{13} back into the original equation. Calculate both sides: Left side: -3(4(-8/13)+8) = -72/13. Right side: 27(-8/13) = -216/13 = -72/13. They match!

Why do I need to simplify the fraction at the end?

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Simplifying gives the cleanest form of your answer. 2439 -\frac{24}{39} and 813 -\frac{8}{13} are equal, but the simplified version is easier to work with in future problems.

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