Let's break down each statement and write the corresponding equation, and then solve each equation step-by-step for the respective variable.
(i) The sum of numbers [tex]\(x\)[/tex] and 4 is 9.
[tex]\[
x + 4 = 9
\][/tex]
Subtract 4 from both sides:
[tex]\[
x = 9 - 4 = 5
\][/tex]
(ii) 2 subtracted from [tex]\(y\)[/tex] is 8.
[tex]\[
y - 2 = 8
\][/tex]
Add 2 to both sides:
[tex]\[
y = 8 + 2 = 10
\][/tex]
(iii) Ten times [tex]\(a\)[/tex] is 70.
[tex]\[
10a = 70
\][/tex]
Divide both sides by 10:
[tex]\[
a = \frac{70}{10} = 7.0
\][/tex]
(iv) The number [tex]\(b\)[/tex] divided by 5 gives 6.
[tex]\[
\frac{b}{5} = 6
\][/tex]
Multiply both sides by 5:
[tex]\[
b = 6 \times 5 = 30
\][/tex]
(v) Three-fourth of [tex]\(t\)[/tex] is 15.
[tex]\[
\frac{3}{4}t = 15
\][/tex]
Multiply both sides by [tex]\(\frac{4}{3}\)[/tex]:
[tex]\[
t = 15 \times \frac{4}{3} = 20.0
\][/tex]
(vi) Seven times [tex]\(m\)[/tex] plus 7 gets you 77.
[tex]\[
7m + 7 = 77
\][/tex]
Subtract 7 from both sides:
[tex]\[
7m = 77 - 7 = 70
\][/tex]
Divide both sides by 7:
[tex]\[
m = \frac{70}{7} = 10.0
\][/tex]
(vii) One-fourth of a number [tex]\(x\)[/tex] minus 4 gives 4.
[tex]\[
\frac{1}{4}x - 4 = 4
\][/tex]
Add 4 to both sides:
[tex]\[
\frac{1}{4}x = 4 + 4 = 8
\][/tex]
Multiply both sides by 4:
[tex]\[
x = 8 \times 4 = 32
\][/tex]
(viii) If you take away 6 from 6 times [tex]\(y\)[/tex], you get 60.
[tex]\[
6y - 6 = 60
\][/tex]
Add 6 to both sides:
[tex]\[
6y = 60 + 6 = 66
\][/tex]
Divide both sides by 6:
[tex]\[
y = \frac{66}{6} = 11.0
\][/tex]
(ix) If you add 3 to one-third of [tex]\(z\)[/tex], you get 30.
[tex]\[
\frac{1}{3}z + 3 = 30
\][/tex]
Subtract 3 from both sides:
[tex]\[
\frac{1}{3}z = 30 - 3 = 27
\][/tex]
Multiply both sides by 3:
[tex]\[
z = 27 \times 3 = 81
\][/tex]
Thus, the solutions are:
[tex]\[
(x_1 = 5, y_1 = 10, a = 7.0, b = 30, t = 20.0, m = 10.0, x_2 = 32, y_2 = 11.0, z = 81)
\][/tex]
