Question

1. solve each of the following equations:

a) $3x + 4 = 25$
b) $5 - 2n = -15$
c) $6a - 3a + 5 = 14$
d) $5x^2 = 4x^2 + 4$
e) $4(2x - 2) = -16$
f) $3(n - 2) - 19 = 5 + 2(n + 5)$
g) $3(4k - 1) - (6k - 10) = 7k$
h) $\frac{x}{4} = \frac{x}{2} + 1$
i) $\frac{3a}{4} - \frac{2a}{3} = \frac{5}{6} + a$
j) $\frac{d - 2}{4} = \frac{d + 1}{3}$
k) $\frac{1}{3}(x + 4) = \frac{1}{5}(x + 2)$
l) $\frac{2x + 1}{4} - \frac{3}{5} = \frac{4x + 7}{2}$
m) $\frac{2}{5}(x + 4) + \frac{3}{2}(x + 2) = -3$

Explanation:

Part a)

Step1: Isolate the x-term

Subtract 4 from both sides.
$3x + 4 - 4 = 25 - 4$
$3x = 21$

Step2: Solve for x

Divide both sides by 3.
$x = \frac{21}{3} = 7$

Part b)

Step1: Isolate the n-term

Subtract 5 from both sides.
$5 - 2n - 5 = -15 - 5$
$-2n = -20$

Step2: Solve for n

Divide both sides by -2.
$n = \frac{-20}{-2} = 10$

Part c)

Step1: Simplify like terms

Combine the a-terms.
$3a + 5 = 14$

Step2: Isolate the a-term

Subtract 5 from both sides.
$3a = 14 - 5 = 9$

Step3: Solve for a

Divide both sides by 3.
$a = \frac{9}{3} = 3$

Part d)

Step1: Simplify the quadratic

Subtract $4x^2$ from both sides.
$5x^2 - 4x^2 = 4$
$x^2 = 4$

Step2: Solve for x

Take square roots of both sides.
$x = \pm\sqrt{4} = \pm2$

Part e)

Step1: Expand the left side

Distribute the 4.
$8x - 8 = -16$

Step2: Isolate the x-term

Add 8 to both sides.
$8x = -16 + 8 = -8$

Step3: Solve for x

Divide both sides by 8.
$x = \frac{-8}{8} = -1$

Part f)

Step1: Expand both sides

Distribute constants to binomials.
$3n - 6 - 19 = 5 + 2n + 10$

Step2: Simplify both sides

Combine constant terms.
$3n - 25 = 2n + 15$

Step3: Isolate the n-term

Subtract $2n$ and add 25 to both sides.
$3n - 2n = 15 + 25$
$n = 40$

Part g)

Step1: Expand both sides

Distribute constants to binomials.
$12k - 3 - 6k + 10 = 7k$

Step2: Simplify left side

Combine like terms.
$6k + 7 = 7k$

Step3: Isolate the k-term

Subtract $6k$ from both sides.
$7 = 7k - 6k$
$k = 7$

Part h)

Step1: Eliminate denominators

Multiply all terms by 4.
$x = 2x + 4$

Step2: Isolate the x-term

Subtract $x$ and 4 from both sides.
$-4 = 2x - x$
$x = -4$

Part i)

Step1: Eliminate denominators

Multiply all terms by 12 (LCM of 4,3,6).
$9a - 8a = 10 + 12a$

Step2: Simplify left side

Combine like terms.
$a = 10 + 12a$

Step3: Isolate the a-term

Subtract $12a$ from both sides.
$a - 12a = 10$
$-11a = 10$

Step4: Solve for a

Divide both sides by -11.
$a = -\frac{10}{11}$

Part j)

Step1: Cross-multiply to eliminate denominators

Multiply both sides by 12 (LCM of 4,3).
$3(d - 2) = 4(d + 1)$

Step2: Expand both sides

Distribute constants.
$3d - 6 = 4d + 4$

Step3: Isolate the d-term

Subtract $3d$ and 4 from both sides.
$-6 - 4 = 4d - 3d$
$d = -10$

Part k)

Step1: Cross-multiply to eliminate denominators

Multiply both sides by 15 (LCM of 3,5).
$5(x + 4) = 3(x + 2)$

Step2: Expand both sides

Distribute constants.
$5x + 20 = 3x + 6$

Step3: Isolate the x-term

Subtract $3x$ and 20 from both sides.
$5x - 3x = 6 - 20$
$2x = -14$

Step4: Solve for x

Divide both sides by 2.
$x = \frac{-14}{2} = -7$

Part l)

Step1: Eliminate denominators

Multiply all terms by 20 (LCM of 4,5,2).
$5(2x + 1) - 12 = 10(4x + 7)$

Step2: Expand both sides

Distribute constants.
$10x + 5 - 12 = 40x + 70$

Step3: Simplify left side

Combine constants.
$10x - 7 = 40x + 70$

Step4: Isolate the x-term

Subtract $10x$ and 70 from both sides.
$-7 - 70 = 40x - 10x$
$30x = -77$

Step5: Solve for x

Divide both sides by 30.
$x = -\frac{77}{30}$

Part m)

Step1: Eliminate denominators

Multiply all terms by 10 (LCM of 5,2).
$4(x + 4) + 15(x + 2) = -30$

Step2: Expand both sides

Distribute constants.
$4x + 16 + 15x + 30 = -30$

Step3: Simplify left side

Combine like terms.
$19x + 46 = -30$

Step4: Isolate the x-term

Subtract 46 from both sides.
$19x = -30 - 46 = -76$

Step5: Solve for x

Divide both sides by 19.
$x = \frac{-76}{19} = -4$

Answer:

a) $x=7$
b) $n=10$
c) $a=3$
d) $x=2$ or $x=-2$
e) $x=-1$
f) $n=40$
g) $k=7$
h) $x=-4$
i) $a=-\frac{10}{11}$
j) $d=-10$
k) $x=-7$
l) $x=-\frac{77}{30}$
m) $x=-4$