Answer :
Certainly! Let's solve the given question step-by-step.
### Part 1: Solving Equations using Trial and Error Method
#### Equation (i): [tex]\(4p + 1 = 37\)[/tex]
To find the value of [tex]\(p\)[/tex], we need to determine a number that satisfies the equation.
1. Start with [tex]\(p = 0\)[/tex]: [tex]\(4(0) + 1 = 1\)[/tex]
2. Try [tex]\(p = 5\)[/tex]: [tex]\(4(5) + 1 = 20 + 1 = 21\)[/tex]
3. Continue with [tex]\(p = 9\)[/tex]: [tex]\(4(9) + 1 = 36 + 1 = 37\)[/tex]
Thus, [tex]\(p = 9\)[/tex] is the solution for the equation [tex]\(4p + 1 = 37\)[/tex].
#### Equation (ii): [tex]\(2p + 1 = 23\)[/tex]
To find the value of [tex]\(p\)[/tex], we need to determine a number that satisfies the equation.
1. Start with [tex]\(p = 0\)[/tex]: [tex]\(2(0) + 1 = 1\)[/tex]
2. Try [tex]\(p = 10\)[/tex]: [tex]\(2(10) + 1 = 20 + 1 = 21\)[/tex]
3. Continue with [tex]\(p = 11\)[/tex]: [tex]\(2(11) + 1 = 22 + 1 = 23\)[/tex]
Thus, [tex]\(p = 11\)[/tex] is the solution for the equation [tex]\(2p + 1 = 23\)[/tex].
#### Equation (iii): [tex]\(7n + 5 = 19\)[/tex]
To find the value of [tex]\(n\)[/tex], we need to determine a number that satisfies the equation.
1. Start with [tex]\(n = 0\)[/tex]: [tex]\(7(0) + 5 = 5\)[/tex]
2. Try [tex]\(n = 1\)[/tex]: [tex]\(7(1) + 5 = 7 + 5 = 12\)[/tex]
3. Continue with [tex]\(n = 2\)[/tex]: [tex]\(7(2) + 5 = 14 + 5 = 19\)[/tex]
Thus, [tex]\(n = 2\)[/tex] is the solution for the equation [tex]\(7n + 5 = 19\)[/tex].
### Part 2: Stating the Inverse of Each Given Operation
#### Operation (i): [tex]\(+5\)[/tex]
The inverse operation of adding 5 is subtracting 5.
Inverse: [tex]\(-5\)[/tex]
#### Operation (v): [tex]\(-\frac{1}{3}\)[/tex]
The inverse operation of subtracting [tex]\(\frac{1}{3}\)[/tex] is adding [tex]\(\frac{1}{3}\)[/tex].
However, interpreting [tex]\(-\frac{1}{3}\)[/tex] as multiplying by [tex]\(-\frac{1}{3}\)[/tex],
Inverse: Multiply by [tex]\(-3\)[/tex].
#### Operation (ii): [tex]\(\times 3\)[/tex]
The inverse operation of multiplication by 3 is division by 3.
Inverse: [tex]\(\div 3\)[/tex] or [tex]\(/\ 3\)[/tex]
#### Operation (iii): [tex]\(\div 2\)[/tex]
The inverse operation of division by 2 is multiplication by 2.
Inverse: [tex]\(\times 2\)[/tex]
#### Operation (vi): [tex]\(\div \frac{1}{4}\)[/tex]
The inverse operation of division by [tex]\(\frac{1}{4}\)[/tex] is multiplication by 4.
Inverse: [tex]\(\times 4\)[/tex]
#### Operation (vii): [tex]\(+\frac{3}{4}\)[/tex]
The inverse operation of adding [tex]\(\frac{3}{4}\)[/tex] is subtracting [tex]\(\frac{3}{4}\)[/tex].
Inverse: [tex]\(-\frac{3}{4}\)[/tex]
### Summary
- Solutions for the equations:
1. [tex]\(p = 9\)[/tex] for [tex]\(4p + 1 = 37\)[/tex]
2. [tex]\(p = 11\)[/tex] for [tex]\(2p + 1 = 23\)[/tex]
3. [tex]\(n = 2\)[/tex] for [tex]\(7n + 5 = 19\)[/tex]
- Inverses of the operations:
1. [tex]\(\text{Inverse of } +5 = -5\)[/tex]
2. [tex]\(\text{Inverse of } -\frac{1}{3} = \times -3\)[/tex]
3. [tex]\(\text{Inverse of } \times 3 = \div 3\)[/tex]
4. [tex]\(\text{Inverse of } \div 2 = \times 2\)[/tex]
5. [tex]\(\text{Inverse of } \div \frac{1}{4} = \times 4\)[/tex]
6. [tex]\(\text{Inverse of } +\frac{3}{4} = -\frac{3}{4}\)[/tex]
I hope this detailed solution helps you understand the steps to solve the given problems!
### Part 1: Solving Equations using Trial and Error Method
#### Equation (i): [tex]\(4p + 1 = 37\)[/tex]
To find the value of [tex]\(p\)[/tex], we need to determine a number that satisfies the equation.
1. Start with [tex]\(p = 0\)[/tex]: [tex]\(4(0) + 1 = 1\)[/tex]
2. Try [tex]\(p = 5\)[/tex]: [tex]\(4(5) + 1 = 20 + 1 = 21\)[/tex]
3. Continue with [tex]\(p = 9\)[/tex]: [tex]\(4(9) + 1 = 36 + 1 = 37\)[/tex]
Thus, [tex]\(p = 9\)[/tex] is the solution for the equation [tex]\(4p + 1 = 37\)[/tex].
#### Equation (ii): [tex]\(2p + 1 = 23\)[/tex]
To find the value of [tex]\(p\)[/tex], we need to determine a number that satisfies the equation.
1. Start with [tex]\(p = 0\)[/tex]: [tex]\(2(0) + 1 = 1\)[/tex]
2. Try [tex]\(p = 10\)[/tex]: [tex]\(2(10) + 1 = 20 + 1 = 21\)[/tex]
3. Continue with [tex]\(p = 11\)[/tex]: [tex]\(2(11) + 1 = 22 + 1 = 23\)[/tex]
Thus, [tex]\(p = 11\)[/tex] is the solution for the equation [tex]\(2p + 1 = 23\)[/tex].
#### Equation (iii): [tex]\(7n + 5 = 19\)[/tex]
To find the value of [tex]\(n\)[/tex], we need to determine a number that satisfies the equation.
1. Start with [tex]\(n = 0\)[/tex]: [tex]\(7(0) + 5 = 5\)[/tex]
2. Try [tex]\(n = 1\)[/tex]: [tex]\(7(1) + 5 = 7 + 5 = 12\)[/tex]
3. Continue with [tex]\(n = 2\)[/tex]: [tex]\(7(2) + 5 = 14 + 5 = 19\)[/tex]
Thus, [tex]\(n = 2\)[/tex] is the solution for the equation [tex]\(7n + 5 = 19\)[/tex].
### Part 2: Stating the Inverse of Each Given Operation
#### Operation (i): [tex]\(+5\)[/tex]
The inverse operation of adding 5 is subtracting 5.
Inverse: [tex]\(-5\)[/tex]
#### Operation (v): [tex]\(-\frac{1}{3}\)[/tex]
The inverse operation of subtracting [tex]\(\frac{1}{3}\)[/tex] is adding [tex]\(\frac{1}{3}\)[/tex].
However, interpreting [tex]\(-\frac{1}{3}\)[/tex] as multiplying by [tex]\(-\frac{1}{3}\)[/tex],
Inverse: Multiply by [tex]\(-3\)[/tex].
#### Operation (ii): [tex]\(\times 3\)[/tex]
The inverse operation of multiplication by 3 is division by 3.
Inverse: [tex]\(\div 3\)[/tex] or [tex]\(/\ 3\)[/tex]
#### Operation (iii): [tex]\(\div 2\)[/tex]
The inverse operation of division by 2 is multiplication by 2.
Inverse: [tex]\(\times 2\)[/tex]
#### Operation (vi): [tex]\(\div \frac{1}{4}\)[/tex]
The inverse operation of division by [tex]\(\frac{1}{4}\)[/tex] is multiplication by 4.
Inverse: [tex]\(\times 4\)[/tex]
#### Operation (vii): [tex]\(+\frac{3}{4}\)[/tex]
The inverse operation of adding [tex]\(\frac{3}{4}\)[/tex] is subtracting [tex]\(\frac{3}{4}\)[/tex].
Inverse: [tex]\(-\frac{3}{4}\)[/tex]
### Summary
- Solutions for the equations:
1. [tex]\(p = 9\)[/tex] for [tex]\(4p + 1 = 37\)[/tex]
2. [tex]\(p = 11\)[/tex] for [tex]\(2p + 1 = 23\)[/tex]
3. [tex]\(n = 2\)[/tex] for [tex]\(7n + 5 = 19\)[/tex]
- Inverses of the operations:
1. [tex]\(\text{Inverse of } +5 = -5\)[/tex]
2. [tex]\(\text{Inverse of } -\frac{1}{3} = \times -3\)[/tex]
3. [tex]\(\text{Inverse of } \times 3 = \div 3\)[/tex]
4. [tex]\(\text{Inverse of } \div 2 = \times 2\)[/tex]
5. [tex]\(\text{Inverse of } \div \frac{1}{4} = \times 4\)[/tex]
6. [tex]\(\text{Inverse of } +\frac{3}{4} = -\frac{3}{4}\)[/tex]
I hope this detailed solution helps you understand the steps to solve the given problems!
