Question

for problems 3-9, solve the equation. check your solution.

1. $\frac{1}{4}x + 12 = x - 6$
2. $1.2a + 10.8 = 54 - 2.4a$
3. $n - 9 = \frac{3}{5}n$
4. $31.5 - 0.5p = 9.5p + 1.5 - 7p$
5. $\frac{2}{3}(5c - 12) = -10 - \frac{1}{6}c$
6. $1.4 - 0.8w = 0.1(1 - 21w)$
7. $\frac{1}{8}(4 + 2b) = \frac{4}{5}(2b + 4)$

Explanation:

Problem 3

Step1: Isolate x terms

Subtract $\frac{1}{4}x$ and add 6 to both sides.
$12 + 6 = x - \frac{1}{4}x$

Step2: Simplify both sides

Combine like terms and solve for x.
$18 = \frac{3}{4}x$
$x = 18 \times \frac{4}{3} = 24$

Step3: Verify solution

Substitute $x=24$ into original equation.
$\frac{1}{4}(24) + 12 = 6 + 12 = 18$; $24 - 6 = 18$

Problem 4

Step1: Isolate a terms

Add $2.4a$ and subtract 10.8 to both sides.
$1.2a + 2.4a = 54 - 10.8$

Step2: Simplify and solve for a

Combine like terms and divide.
$3.6a = 43.2$
$a = \frac{43.2}{3.6} = 12$

Step3: Verify solution

Substitute $a=12$ into original equation.
$1.2(12)+10.8=14.4+10.8=25.2$; $54-2.4(12)=54-28.8=25.2$

Problem 5

Step1: Isolate n terms

Subtract $\frac{3}{5}n$ and add 9 to both sides.
$n - \frac{3}{5}n = 9$

Step2: Simplify and solve for n

Combine like terms and solve.
$\frac{2}{5}n = 9$
$n = 9 \times \frac{5}{2} = 22.5$

Step3: Verify solution

Substitute $n=22.5$ into original equation.
$22.5 - 9 = 13.5$; $\frac{3}{5}(22.5)=13.5$

Problem 6

Step1: Simplify right-hand side

Combine like p terms.
$31.5 - 0.5p = 2.5p + 1.5$

Step2: Isolate p terms

Add $0.5p$ and subtract 1.5 to both sides.
$31.5 - 1.5 = 2.5p + 0.5p$

Step3: Solve for p

Simplify and divide.
$30 = 3p$
$p = 10$

Step4: Verify solution

Substitute $p=10$ into original equation.
$31.5-0.5(10)=26.5$; $9.5(10)+1.5-7(10)=95+1.5-70=26.5$

Problem 7

Step1: Eliminate denominators

Multiply all terms by 6 (LCM of 3,6).
$4(5c - 12) = -60 - c$

Step2: Expand left-hand side

Distribute the 4.
$20c - 48 = -60 - c$

Step3: Isolate c terms

Add c and 48 to both sides.
$20c + c = -60 + 48$

Step4: Solve for c

Simplify and divide.
$21c = -12$
$c = -\frac{12}{21} = -\frac{4}{7}$

Step5: Verify solution

Substitute $c=-\frac{4}{7}$ into original equation.
$\frac{2}{3}(5(-\frac{4}{7})-12)=\frac{2}{3}(-\frac{20}{7}-\frac{84}{7})=\frac{2}{3}(-\frac{104}{7})=-\frac{208}{21}$;
$-10 - \frac{1}{6}(-\frac{4}{7})=-10+\frac{2}{21}=-\frac{210}{21}+\frac{2}{21}=-\frac{208}{21}$

Problem 8

Step1: Expand right-hand side

Distribute the 0.1.
$1.4 - 0.8w = 0.1 - 2.1w$

Step2: Isolate w terms

Add $2.1w$ and subtract 1.4 to both sides.
$-0.8w + 2.1w = 0.1 - 1.4$

Step3: Solve for w

Simplify and divide.
$1.3w = -1.3$
$w = -1$

Step4: Verify solution

Substitute $w=-1$ into original equation.
$1.4-0.8(-1)=1.4+0.8=2.2$; $0.1(1-21(-1))=0.1(22)=2.2$

Problem 9

Step1: Eliminate denominators

Multiply all terms by 40 (LCM of 8,5).
$5(4 + 2b) = 32(2b + 4)$

Step2: Expand both sides

Distribute constants.
$20 + 10b = 64b + 128$

Step3: Isolate b terms

Subtract $10b$ and 128 from both sides.
$20 - 128 = 64b - 10b$

Step4: Solve for b

Simplify and divide.
$-108 = 54b$
$b = -2$

Step5: Verify solution

Substitute $b=-2$ into original equation.
$\frac{1}{8}(4+2(-2))=\frac{1}{8}(0)=0$; $\frac{4}{5}(2(-2)+4)=\frac{4}{5}(0)=0$

Answer:

1. $x=24$
2. $a=12$
3. $n=22.5$
4. $p=10$
5. $c=-\frac{4}{7}$
6. $w=-1$
7. $b=-2$