Simplify 54x+38y-3(4x-7y): Linear Expression with Distribution

Distribution with Negative Multiplier

54x+38y3(4x7y)=? 54x+38y-3(4x-7y)=\text{?}

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Step-by-step video solution

Watch the teacher solve the problem with clear explanations
00:00 Solve
00:06 Note, when multiplying negative by positive it's always negative
00:13 Note, when multiplying negative by negative it's always positive
00:30 Let's solve the multiplications
00:44 Let's group the factors
00:52 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

54x+38y3(4x7y)=? 54x+38y-3(4x-7y)=\text{?}

2

Step-by-step solution

We begin by addressing the parenthesis:

Remember that:

When we multiply a positive number by a negative number, the result will be negative.

When we multiply a negative number by a negative number, the result will be positive.

Hence we obtain the following:

54x+38y3×4x3×(7y)= 54x+38y-3\times4x-3\times(-7y)=

We then multiply and change the sign according to the rules:

54x+38y12x+21y= 54x+38y-12x+21y=

Next we join together the x coefficients:

54x12x=42x 54x-12x=42x

We also join together the y coefficients:

38y+21y=59y 38y+21y=59y

Finally we obtain:

42x+59y 42x+59y

3

Final Answer

42x+59y 42x+59y

Key Points to Remember

Essential concepts to master this topic
  • Distribution Rule: Multiply each term inside parentheses by outside factor
  • Technique: -3(4x-7y) becomes -3×4x and -3×(-7y) = -12x+21y
  • Check: Combine like terms correctly: 54x-12x=42x and 38y+21y=59y ✓

Common Mistakes

Avoid these frequent errors
  • Forgetting to distribute the negative sign to all terms
    Don't just multiply -3 by the first term getting -12x and ignore the sign rules for -7y = wrong signs throughout! This creates incorrect coefficients and wrong final answers. Always multiply the outside factor by every single term inside the parentheses, paying careful attention to positive and negative signs.

Practice Quiz

Test your knowledge with interactive questions

\( 100-(5+55)= \)

FAQ

Everything you need to know about this question

Why does -3 times -7y become +21y?

+

Remember the sign rules: negative × negative = positive! So -3 × (-7y) = +21y. This is why we get +21y instead of -21y in our final expression.

How do I know which terms to combine?

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Only combine like terms - terms with the same variable and exponent. Here, combine all x terms together (54x - 12x) and all y terms together (38y + 21y).

What if I get confused with the negative signs?

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Write out each step clearly! Show -3(4x-7y) = -3×4x + (-3)×(-7y) = -12x + 21y. Taking your time with signs prevents most errors.

Can I distribute in a different order?

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Yes! You can distribute to either term first, but make sure you distribute to both terms. The final result will be the same as long as you follow the sign rules correctly.

How do I check if my final answer is right?

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Substitute simple values for x and y into both the original expression and your simplified answer. If they give the same result, you're correct! For example, try x=1, y=1.

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