Solve for X: Rectangle with Area 78 cm² and Width 3 cm

Rectangle Area with Algebraic Expressions

Given the rectangular area 78 cm².

Find X

S=78S=78S=78X+7X+7X+7333

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1

Understand the problem

Given the rectangular area 78 cm².

Find X

S=78S=78S=78X+7X+7X+7333

2

Step-by-step solution

We know that the area of a rectangle is equal to its length multiplied by its width.

We begin by writing an equation with the available data.

78=3×(x+7) 78=3\times(x+7)

We then use the distributive property to solve the equation.

That is, we multiply each of the terms inside of the parentheses by 3:

78=3×x+3×7 78=3\times x+3\times7

78=3x+21 78=3x+21

We move 21 to the other side and use the appropriate sign:

7821=3x 78-21=3x

57=3x 57=3x

Lastly we divide both sides by 3:

573=3x3 \frac{57}{3}=\frac{3x}{3}

x=19 x=19

3

Final Answer

19 19

Key Points to Remember

Essential concepts to master this topic
  • Formula: Area of rectangle equals length times width
  • Technique: Set up equation: 78=3×(x+7) 78 = 3 \times (x + 7)
  • Check: Substitute x = 19: 3×26=78 3 \times 26 = 78

Common Mistakes

Avoid these frequent errors
  • Forgetting to apply distributive property correctly
    Don't just divide 78 by 3 to get x = 26! This ignores the +7 inside parentheses and gives wrong results. Always distribute first: 3(x+7)=3x+21 3(x + 7) = 3x + 21 , then solve the resulting equation.

Practice Quiz

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FAQ

Everything you need to know about this question

Why can't I just divide 78 by 3 to find x?

+

Because the width is 3 and the length is (x + 7), not just x! You need to set up the equation 78=3×(x+7) 78 = 3 \times (x + 7) first, then solve step by step.

What does the distributive property do here?

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The distributive property helps us expand 3(x+7) 3(x + 7) into 3x+21 3x + 21 . This removes the parentheses so we can solve the linear equation normally.

How do I know which side is length and which is width?

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It doesn't matter! In rectangles, area = length × width works the same as area = width × length. The important thing is identifying which measurement goes with which label.

What if I get a negative answer for x?

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Check your arithmetic! In this problem, x should be positive since it represents part of a length measurement. Negative lengths don't make sense in real-world rectangle problems.

Why do we subtract 21 from both sides?

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We want to isolate the term with x. Since we have 78=3x+21 78 = 3x + 21 , subtracting 21 from both sides gives us 57=3x 57 = 3x , which is easier to solve.

How can I verify my answer is correct?

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Substitute x = 19 back into the original setup: length = 19 + 7 = 26 cm, width = 3 cm. Check: 26×3=78 26 \times 3 = 78 cm² ✓

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