Question

name:
math 7
directions: answer each of the following questions. show all work.

1. the length of a garden is represented by 4x. the width of the garden is represented by 5y. express the perimeter of the garden as a simplified expression.
2. translate: the product of 6 and 21 minus a number
3. solve: \\( \frac { 3 } { 5 } x = - 6 \\)
4. solve: \\( 124 = - 11 x + 3 \\)
5. simplify: \\( - 5 ( 1 + 2 k ) - 8 ( - 4 + 5 k ) \\)
6. factor: \\( 80 x y - 16 x \\)
7. translate: five more than twice a number
8. use your expression from #7 to evaluate for when \\( m = 2.3 \\)
9. compare the following: \\( - 1 \frac { 1 } { 2 } \textcircled { > } - 2 \\)
10. evaluate:

a. \\( \sqrt { 49 } = \\)
b. \\( \sqrt { 8 } = \\)
c. \\( \sqrt { 100 } = \\)
teacher use only:

Explanation:

1. Perimeter of a rectangle formula

The formula for the perimeter \(P\) of a rectangle is \(P = 2(l + w)\), where \(l\) is the length and \(w\) is the width. Given \(l = 4x\) and \(w = 5y\).

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2. Translating the phrase

“The product of 6 and 21” is \(6\times21\), and “minus a number” (let the number be \(n\)) gives \(6\times21 - n=126 - n\).

3. Solving the equation \(\frac{3}{5}x=-6\)

Multiply both sides of the equation by \(\frac{5}{3}\) to isolate \(x\).

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4. Solving the equation \(124=-11x + 3\)

First, subtract 3 from both sides: \(124 - 3=-11x\), so \(121=-11x\). Then divide both sides by \(-11\).

$$ x=\frac{121}{-11}=-11 $$

5. Simplifying \(-5(1 + 2k)-8(-4 + 5k)\)

Use the distributive property \(a(b + c)=ab+ac\).

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6. Factoring \(80xy-16x\)

Find the greatest common factor (GCF) of \(80xy\) and \(16x\). The GCF of 80 and 16 is 16, and the common variable factor is \(x\).

$$80xy-16x = 16x(5y - 1)$$

7. Translating “Five more than twice a number”

Let the number be \(m\). “Twice a number” is \(2m\), and “five more than” gives \(2m+5\).

9. Comparing \(-1\frac{1}{2}\) and \(-2\)

Convert \(-1\frac{1}{2}=-\frac{3}{2}=-1.5\). Since \(-1.5>-2\), we have \(-1\frac{1}{2}>-2\).

10. Evaluating square - roots

• For \(a.\), since \(7\times7 = 49\), \(\sqrt{49}=7\).
• For \(b.\), \(\sqrt{8}=\sqrt{4\times2}=2\sqrt{2}\approx2.83\).
• For \(c.\), since \(10\times10 = 100\), \(\sqrt{100}=10\).

Answer:

1. \(8x + 10y\)
2. \(126 - n\)
3. \(x=-10\)
4. \(x = - 11\)
5. \(-50k + 27\)
6. \(16x(5y - 1)\)
7. \(2m + 5\)
8. \(-1\frac{1}{2}>-2\)

10.

• a. \(7\)
• b. \(2\sqrt{2}\approx2.83\)
• c. \(10\)