Solve for X in 3(x+2)=5(2-x): Linear Equation with Distributive Property

Linear Equations with Distribution and Collection

Solve for X:


3(x+2)=5(2−x) 3(x+2)=5(2-x)

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Find X
00:03 Make sure to open brackets properly, multiply by each factor
00:21 Solve each multiplication separately
00:25 Positive times negative is always negative
00:29 Arrange the equation so that X is isolated on one side
00:46 Collect like terms
00:56 Isolate X
01:09 Simplify as much as possible
01:12 Break down 8 into factors 4 and 2
01:15 Simplify as much as possible
01:18 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 for X:


3(x+2)=5(2−x) 3(x+2)=5(2-x)

2

Step-by-step solution

To solve the given equation 3(x+2)=5(2−x)3(x+2) = 5(2-x), we will follow these steps:

  • Step 1: Distribute on both sides of the equation.

For the left side, distribute 33 over (x+2)(x+2):

3(x+2)=3x+63(x+2) = 3x + 6

For the right side, distribute 55 over (2−x)(2-x):

5(2−x)=10−5x5(2-x) = 10 - 5x

  • Step 2: Rewrite the equation with the expanded terms.

The equation becomes:

3x+6=10−5x3x + 6 = 10 - 5x

  • Step 3: Combine like terms to isolate xx.

First, add 5x5x to both sides to get all xx terms on the left side:

3x+5x+6=103x + 5x + 6 = 10

This simplifies to:

8x+6=108x + 6 = 10

Next, subtract 6 from both sides to isolate the term with xx:

8x=48x = 4

  • Step 4: Solve for xx.

Divide both sides by 8 to solve for xx:

x=48x = \frac{4}{8}

Simplify the fraction:

x=12x = \frac{1}{2}

Therefore, the solution to the equation is x=12x = \frac{1}{2}.

3

Final Answer

x=12 x=\frac{1}{2}

Key Points to Remember

Essential concepts to master this topic
  • Distribution Rule: Apply each factor to every term inside parentheses
  • Collection Technique: Move all x-terms to one side: 3x + 5x = 8x
  • Verification: Substitute x=12 x = \frac{1}{2} back into original equation: both sides equal 152 \frac{15}{2} ✓

Common Mistakes

Avoid these frequent errors
  • Incorrectly distributing negative signs
    Don't distribute 5 over (2-x) as 10 + 5x = wrong signs! This makes you combine terms incorrectly and get x = -8 instead of x = 1/2. Always distribute carefully: 5(2-x) = 10 - 5x, keeping the negative with 5x.

Practice Quiz

Test your knowledge with interactive questions

Solve for X:

\( x - 3 + 5 = 8 - 2 \)

FAQ

Everything you need to know about this question

Why do I need to distribute first instead of solving differently?

+

The distributive property is required when you have parentheses. You cannot move terms or combine anything until you've expanded both sides completely!

How do I handle the negative sign in 5(2-x)?

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Be extra careful! The 5 multiplies both terms: 5×2=10 5 \times 2 = 10 and 5×(−x)=−5x 5 \times (-x) = -5x . So 5(2-x) = 10 - 5x, not 10 + 5x.

What if I get a fraction as my answer?

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Fractions are completely normal answers! x=12 x = \frac{1}{2} is perfectly valid. Just make sure to simplify and always check by substituting back into the original equation.

Why do I move all x-terms to one side?

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This helps you combine like terms efficiently. When you have 3x on the left and -5x on the right, moving them together gives you 8x = 4, which is much easier to solve!

How can I check if my answer is correct?

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Substitute x=12 x = \frac{1}{2} into the original equation:

  • Left side: 3(12+2)=3(52)=152 3(\frac{1}{2}+2) = 3(\frac{5}{2}) = \frac{15}{2}
  • Right side: 5(2−12)=5(32)=152 5(2-\frac{1}{2}) = 5(\frac{3}{2}) = \frac{15}{2}

Both sides equal 152 \frac{15}{2} , so you're correct!

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