Certainly! Let’s solve each equation step by step using inverse operations.
### Part (a): [tex]\(2f + 18 = 114\)[/tex]
1. Isolate the term with the variable [tex]\(f\)[/tex]:
- Subtract 18 from both sides of the equation:
[tex]\[
2f + 18 - 18 = 114 - 18
\][/tex]
This simplifies to:
[tex]\[
2f = 96
\][/tex]
2. Solve for [tex]\(f\)[/tex]:
- Divide both sides by 2:
[tex]\[
\frac{2f}{2} = \frac{96}{2}
\][/tex]
This simplifies to:
[tex]\[
f = 48
\][/tex]
So the solution for part (a) is [tex]\(f = 48\)[/tex].
### Part (b): [tex]\(tt - 19 = 100\)[/tex]
Here, "tt" is understood to mean [tex]\(t^2\)[/tex] (t squared).
1. Isolate the term with the variable [tex]\(t\)[/tex]:
- Add 19 to both sides of the equation:
[tex]\[
t^2 - 19 + 19 = 100 + 19
\][/tex]
This simplifies to:
[tex]\[
t^2 = 119
\][/tex]
2. Solve for [tex]\(t\)[/tex]:
- Take the square root of both sides:
[tex]\[
t = \sqrt{119} \quad \text{or} \quad t = -\sqrt{119}
\][/tex]
So the solutions for part (b) are [tex]\(t = \sqrt{119}\)[/tex] and [tex]\(t = -\sqrt{119}\)[/tex], which approximate to [tex]\(10.908712114635714\)[/tex] and [tex]\(-10.908712114635714\)[/tex].
### Part (c): [tex]\(200 + 9 / = 119\)[/tex]
This equation appears to be ambiguous, but we'll assume it's a misunderstanding and requires additional context. Hence, no conclusive solution can be determined from it.
### Part (d): [tex]\(106 = 16 + 6m\)[/tex]
1. Isolate the term with the variable [tex]\(m\)[/tex]:
- Subtract 16 from both sides of the equation:
[tex]\[
106 - 16 = 16 + 6m - 16
\][/tex]
This simplifies to:
[tex]\[
90 = 6m
\][/tex]
2. Solve for [tex]\(m\)[/tex]:
- Divide both sides by 6:
[tex]\[
\frac{90}{6} = \frac{6m}{6}
\][/tex]
This simplifies to:
[tex]\[
m = 15
\][/tex]
So the solution for part (d) is [tex]\(m = 15\)[/tex].
