Question

31.
7x = 21 - x
+ x + x
8x = 21

32.
4 - 20 = 16x

33.
10 - 9 = 16 - 5

34.
4(x + 1) = 4 - 1

36.
5(8 - 3) = 9 + 2

linear equations practice
36.
(some handwritten steps and equations, e.g., x - \frac{5}{4} = 2.8

1. \frac{x}{4} - \frac{x}{6} = \frac{1}{4}
2. 9 - \frac{3}{5} + \frac{9}{5} = 20)

Explanation:

Step1: Solve 31: Isolate $x$

$7x = 21 \implies x = \frac{21}{7}$

Step2: Solve 32: Isolate $x$

$4 - 20 = 16x \implies -16 = 16x \implies x = \frac{-16}{16}$

Step3: Solve 33: Isolate $x$

$10 - 9 = 9x - 5 \implies 1 = 9x - 5 \implies 1+5=9x \implies 6=9x \implies x=\frac{6}{9}$

Step4: Solve 34: Expand and isolate $z$

$4(z+1)=4-1 \implies 4z+4=3 \implies 4z=3-4 \implies 4z=-1 \implies z=\frac{-1}{4}$

Step5: Solve 35: Expand and isolate $a$

$5(a-5)=a-2 \implies 5a-25=a-2 \implies 5a-a=-2+25 \implies 4a=23 \implies a=\frac{23}{4}$

Step6: Solve 36: Isolate $n$

$\frac{10}{n}=9-\frac{3}{4} \implies \frac{10}{n}=\frac{36}{4}-\frac{3}{4}=\frac{33}{4} \implies n=10\times\frac{4}{33}$

Step7: Solve 37: Verify equality

$\frac{1}{8}+\frac{5}{4}=\frac{1}{8}+\frac{10}{8}=\frac{11}{8}
eq \frac{10}{5}=2$

Step8: Solve 38: Isolate $x$

$x-\frac{3}{4}=2x \implies -2x+x=\frac{3}{4} \implies -x=\frac{3}{4} \implies x=-\frac{3}{4}$

Step9: Solve 39: Combine fractions

$\frac{x}{4}-\frac{x}{6}=\frac{1}{14} \implies \frac{3x-2x}{12}=\frac{1}{14} \implies \frac{x}{12}=\frac{1}{14} \implies x=\frac{12}{14}$

Step10: Solve 40: Isolate $a$

$a-\frac{a}{5}+\frac{a}{3}=20 \implies \frac{15a-3a+5a}{15}=20 \implies \frac{17a}{15}=20 \implies a=20\times\frac{15}{17}$

Answer:

1. $x=3$
2. $x=-1$
3. $x=\frac{2}{3}$
4. $z=-\frac{1}{4}$
5. $a=\frac{23}{4}$
6. $n=\frac{40}{33}$
7. False (the equation is not valid)
8. $x=-\frac{3}{4}$
9. $x=\frac{6}{7}$
10. $a=\frac{300}{17}$