Question

skill #2: solve basic equations
solve each equation.

1. $5c - 9 = 8 - 2c$ 14. $5(2 - g) = 0$ 15. $6(n - 4) = 3n$
2. $x^2 = 16$ 17. $3x^2 - 5 = 22$ 18. $sqrt{x} + 3 = 8$
3. 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?
4. the length of a rectangle is 3 cm greater than the width. the perimeter is 24 cm. what are the dimensions of the rectangle?

solve each formula for the indicated variable.

1. $v = s^2 + \frac{1}{2}sh$; for $h$ 22. $c = \frac{5}{9}(f - 32)$; for $f$.

Explanation:

Problem 13:

Step1: Collect like terms

$5c + 2c = 8 + 9$

Step2: Simplify both sides

$7c = 17$

Step3: Isolate $c$

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

Problem 14:

Step1: Divide both sides by 5

$2 - g = 0$

Step2: Isolate $g$

$g = 2$

Problem 15:

Step1: Expand left side

$6n - 24 = 3n$

Step2: Collect like terms

$6n - 3n = 24$

Step3: Simplify and solve

$3n = 24 \implies n = 8$

Problem 16:

Step1: Take square root of both sides

$x = \pm\sqrt{16}$

Step2: Calculate root

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

Problem 17:

Step1: Isolate $x^2$ term

$3x^2 = 22 + 5$

Step2: Simplify right side

$3x^2 = 27$

Step3: Solve for $x^2$

$x^2 = 9$

Step4: Take square root

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

Problem 18:

Step1: Isolate square root term

$\sqrt{x} = 8 - 3$

Step2: Simplify right side

$\sqrt{x} = 5$

Step3: Square both sides

$x = 5^2 = 25$

Problem 19:

Step1: Define variables

Let second stage burn time = $t$ seconds. First stage = $t+28$ seconds.

Step2: Set up total time equation

$t + (t+28) = 158$

Step3: Simplify and solve for $t$

$2t + 28 = 158 \implies 2t = 130 \implies t = 65$

Step4: Find first stage time

$t+28 = 65+28 = 93$

Problem 20:

Step1: Define variables

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

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

$w+3 = 4.5+3 = 7.5$

Problem 21:

Step1: Isolate $h$ term

$v - s^2 = \frac{1}{2}sh$

Step2: Multiply by 2

$2(v - s^2) = sh$

Step3: Isolate $h$

$h = \frac{2(v - s^2)}{s}$

Problem 22:

Step1: Multiply both sides by $\frac{9}{5}$

$\frac{9}{5}C = F - 32$

Step2: Isolate $F$

$F = \frac{9}{5}C + 32$

Answer:

1. $c=\frac{17}{7}$
2. $g=2$
3. $n=8$
4. $x=4$ or $x=-4$
5. $x=3$ or $x=-3$
6. $x=25$
7. First stage: 93 seconds, Second stage: 65 seconds
8. Width: 4.5 cm, Length: 7.5 cm
9. $h=\frac{2(v - s^2)}{s}$
10. $F=\frac{9}{5}C + 32$