Question

skill #2: solve basic equations
solve each equation.

1. $5c - 9 = 8 - 2c$ 14. $5(2 - g) = 0$ 15. $6(n - 4) = 3n$

1. $x^2 = 16$ 17. $3x^2 - 5 = 22$ 18. $sqrt{x} + 3 = 8$

1. the first stage of a rocket burns for 28 seconds longer than the second stage. if the rocket’s total burning time is 158 seconds, for how long does each stage burn?

1. the length of a rectangle is 3 cm greater than the width. the perimeter is 24 cm. what are the dimensions of the rectangle?

Explanation:

Step1: Isolate variable terms

$5c + 2c = 8 + 9$

Step2: Combine like terms

$7c = 17$

Step3: Solve for $c$

$c = \frac{17}{7}$

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Step1: Divide both sides by 5

$2 - g = 0$

Step2: Isolate $g$

$g = 2$

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Step1: Expand left side

$6n - 24 = 3n$

Step2: Isolate variable terms

$6n - 3n = 24$

Step3: Combine like terms & solve

$3n = 24 \implies n = 8$

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Step1: Take square roots

$x = \pm\sqrt{16}$

Step2: Simplify root

$x = 4 \text{ or } x = -4$

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Step1: Isolate squared term

$3x^2 = 22 + 5$

Step2: Simplify right side

$3x^2 = 27$

Step3: Solve for $x^2$

$x^2 = 9$

Step4: Take square roots

$x = 3 \text{ or } x = -3$

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Step1: Isolate square root

$\sqrt{x} = 8 - 3$

Step2: Simplify right side

$\sqrt{x} = 5$

Step3: Square both sides

$x = 5^2 = 25$

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Step1: Define variables

Let $t$ = second stage time. First stage: $t + 28$

Step2: Set total time equation

$t + (t + 28) = 158$

Step3: Simplify equation

$2t + 28 = 158$

Step4: Solve for $t$

$2t = 130 \implies t = 65$

Step5: Find first stage time

$t + 28 = 65 + 28 = 93$

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Step1: Define variables

Let $w$ = width. Length: $w + 3$

Step2: Use perimeter formula

$2(w + (w + 3)) = 24$

Step3: Simplify equation

$2(2w + 3) = 24 \implies 2w + 3 = 12$

Step4: Solve for $w$

$2w = 9 \implies w = 4.5$

Step5: Find length

$w + 3 = 4.5 + 3 = 7.5$

Answer:

1. $\boldsymbol{c = \frac{17}{7}}$
2. $\boldsymbol{g = 2}$
3. $\boldsymbol{n = 8}$
4. $\boldsymbol{x = 4 \text{ or } x = -4}$
5. $\boldsymbol{x = 3 \text{ or } x = -3}$
6. $\boldsymbol{x = 25}$
7. First stage: $\boldsymbol{93}$ seconds; Second stage: $\boldsymbol{65}$ seconds
8. Width: $\boldsymbol{4.5}$ cm; Length: $\boldsymbol{7.5}$ cm