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Sagot :
Sure, let's solve the equation [tex]\(\frac{1}{2} m - \frac{3}{4} n = 16\)[/tex] for different values of [tex]\(n\)[/tex]. We will derive the value of [tex]\(m\)[/tex] in each case.
### 1. When [tex]\(n = 20\)[/tex]
The equation we start with is:
[tex]\[ \frac{1}{2} m - \frac{3}{4} (20) = 16 \][/tex]
First, calculate [tex]\(\frac{3}{4} \times 20\)[/tex]:
[tex]\[ \frac{3}{4} \times 20 = 15 \][/tex]
Now, substitute this back into the equation:
[tex]\[ \frac{1}{2} m - 15 = 16 \][/tex]
To isolate [tex]\(\frac{1}{2} m\)[/tex], add 15 to both sides:
[tex]\[ \frac{1}{2} m = 16 + 15 \][/tex]
[tex]\[ \frac{1}{2} m = 31 \][/tex]
Next, multiply both sides by 2 to solve for [tex]\(m\)[/tex]:
[tex]\[ m = 31 \times 2 \][/tex]
[tex]\[ m = 62 \][/tex]
### 2. When [tex]\(n = 32\)[/tex]
The equation we start with is:
[tex]\[ \frac{1}{2} m - \frac{3}{4} (32) = 16 \][/tex]
First, calculate [tex]\(\frac{3}{4} \times 32\)[/tex]:
[tex]\[ \frac{3}{4} \times 32 = 24 \][/tex]
Now, substitute this back into the equation:
[tex]\[ \frac{1}{2} m - 24 = 16 \][/tex]
To isolate [tex]\(\frac{1}{2} m\)[/tex], add 24 to both sides:
[tex]\[ \frac{1}{2} m = 16 + 24 \][/tex]
[tex]\[ \frac{1}{2} m = 40 \][/tex]
Next, multiply both sides by 2 to solve for [tex]\(m\)[/tex]:
[tex]\[ m = 40 \times 2 \][/tex]
[tex]\[ m = 80 \][/tex]
### 3. When [tex]\(n = 44\)[/tex]
The equation we start with is:
[tex]\[ \frac{1}{2} m - \frac{3}{4} (44) = 16 \][/tex]
First, calculate [tex]\(\frac{3}{4} \times 44\)[/tex]:
[tex]\[ \frac{3}{4} \times 44 = 33 \][/tex]
Now, substitute this back into the equation:
[tex]\[ \frac{1}{2} m - 33 = 16 \][/tex]
To isolate [tex]\(\frac{1}{2} m\)[/tex], add 33 to both sides:
[tex]\[ \frac{1}{2} m = 16 + 33 \][/tex]
[tex]\[ \frac{1}{2} m = 49 \][/tex]
Next, multiply both sides by 2 to solve for [tex]\(m\)[/tex]:
[tex]\[ m = 49 \times 2 \][/tex]
[tex]\[ m = 98 \][/tex]
### 4. When [tex]\(n = 48\)[/tex]
The equation we start with is:
[tex]\[ \frac{1}{2} m - \frac{3}{4} (48) = 16 \][/tex]
First, calculate [tex]\(\frac{3}{4} \times 48\)[/tex]:
[tex]\[ \frac{3}{4} \times 48 = 36 \][/tex]
Now, substitute this back into the equation:
[tex]\[ \frac{1}{2} m - 36 = 16 \][/tex]
To isolate [tex]\(\frac{1}{2} m\)[/tex], add 36 to both sides:
[tex]\[ \frac{1}{2} m = 16 + 36 \][/tex]
[tex]\[ \frac{1}{2} m = 52 \][/tex]
Next, multiply both sides by 2 to solve for [tex]\(m\)[/tex]:
[tex]\[ m = 52 \times 2 \][/tex]
[tex]\[ m = 104 \][/tex]
So, for the given values of [tex]\(n\)[/tex], the corresponding values of [tex]\(m\)[/tex] are:
- When [tex]\(n = 20\)[/tex], [tex]\(m = 62\)[/tex]
- When [tex]\(n = 32\)[/tex], [tex]\(m = 80\)[/tex]
- When [tex]\(n = 44\)[/tex], [tex]\(m = 98\)[/tex]
- When [tex]\(n = 48\)[/tex], [tex]\(m = 104\)[/tex]
### 1. When [tex]\(n = 20\)[/tex]
The equation we start with is:
[tex]\[ \frac{1}{2} m - \frac{3}{4} (20) = 16 \][/tex]
First, calculate [tex]\(\frac{3}{4} \times 20\)[/tex]:
[tex]\[ \frac{3}{4} \times 20 = 15 \][/tex]
Now, substitute this back into the equation:
[tex]\[ \frac{1}{2} m - 15 = 16 \][/tex]
To isolate [tex]\(\frac{1}{2} m\)[/tex], add 15 to both sides:
[tex]\[ \frac{1}{2} m = 16 + 15 \][/tex]
[tex]\[ \frac{1}{2} m = 31 \][/tex]
Next, multiply both sides by 2 to solve for [tex]\(m\)[/tex]:
[tex]\[ m = 31 \times 2 \][/tex]
[tex]\[ m = 62 \][/tex]
### 2. When [tex]\(n = 32\)[/tex]
The equation we start with is:
[tex]\[ \frac{1}{2} m - \frac{3}{4} (32) = 16 \][/tex]
First, calculate [tex]\(\frac{3}{4} \times 32\)[/tex]:
[tex]\[ \frac{3}{4} \times 32 = 24 \][/tex]
Now, substitute this back into the equation:
[tex]\[ \frac{1}{2} m - 24 = 16 \][/tex]
To isolate [tex]\(\frac{1}{2} m\)[/tex], add 24 to both sides:
[tex]\[ \frac{1}{2} m = 16 + 24 \][/tex]
[tex]\[ \frac{1}{2} m = 40 \][/tex]
Next, multiply both sides by 2 to solve for [tex]\(m\)[/tex]:
[tex]\[ m = 40 \times 2 \][/tex]
[tex]\[ m = 80 \][/tex]
### 3. When [tex]\(n = 44\)[/tex]
The equation we start with is:
[tex]\[ \frac{1}{2} m - \frac{3}{4} (44) = 16 \][/tex]
First, calculate [tex]\(\frac{3}{4} \times 44\)[/tex]:
[tex]\[ \frac{3}{4} \times 44 = 33 \][/tex]
Now, substitute this back into the equation:
[tex]\[ \frac{1}{2} m - 33 = 16 \][/tex]
To isolate [tex]\(\frac{1}{2} m\)[/tex], add 33 to both sides:
[tex]\[ \frac{1}{2} m = 16 + 33 \][/tex]
[tex]\[ \frac{1}{2} m = 49 \][/tex]
Next, multiply both sides by 2 to solve for [tex]\(m\)[/tex]:
[tex]\[ m = 49 \times 2 \][/tex]
[tex]\[ m = 98 \][/tex]
### 4. When [tex]\(n = 48\)[/tex]
The equation we start with is:
[tex]\[ \frac{1}{2} m - \frac{3}{4} (48) = 16 \][/tex]
First, calculate [tex]\(\frac{3}{4} \times 48\)[/tex]:
[tex]\[ \frac{3}{4} \times 48 = 36 \][/tex]
Now, substitute this back into the equation:
[tex]\[ \frac{1}{2} m - 36 = 16 \][/tex]
To isolate [tex]\(\frac{1}{2} m\)[/tex], add 36 to both sides:
[tex]\[ \frac{1}{2} m = 16 + 36 \][/tex]
[tex]\[ \frac{1}{2} m = 52 \][/tex]
Next, multiply both sides by 2 to solve for [tex]\(m\)[/tex]:
[tex]\[ m = 52 \times 2 \][/tex]
[tex]\[ m = 104 \][/tex]
So, for the given values of [tex]\(n\)[/tex], the corresponding values of [tex]\(m\)[/tex] are:
- When [tex]\(n = 20\)[/tex], [tex]\(m = 62\)[/tex]
- When [tex]\(n = 32\)[/tex], [tex]\(m = 80\)[/tex]
- When [tex]\(n = 44\)[/tex], [tex]\(m = 98\)[/tex]
- When [tex]\(n = 48\)[/tex], [tex]\(m = 104\)[/tex]
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