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Sagot :
Sure! Let's solve each of these equations step by step.
### 1. Solving [tex]\(\frac{3}{4} x = 9\)[/tex] for [tex]\(x\)[/tex]:
First, isolate [tex]\(x\)[/tex]:
[tex]\[ \frac{3}{4} x = 9 \][/tex]
Multiply both sides by [tex]\(\frac{4}{3}\)[/tex] to get [tex]\(x\)[/tex] alone:
[tex]\[ x = 9 \times \frac{4}{3} \][/tex]
Simplifying the right side:
[tex]\[ x = 12 \][/tex]
### 2. Solving [tex]\(a + x = b\)[/tex] for [tex]\(x\)[/tex]:
We need to express [tex]\(x\)[/tex] in terms of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ a + x = b \][/tex]
Subtract [tex]\(a\)[/tex] from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[ x = b - a \][/tex]
### 3. Solving [tex]\(c x = d\)[/tex] for [tex]\(x\)[/tex]:
We need to express [tex]\(x\)[/tex] in terms of [tex]\(c\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[ c x = d \][/tex]
Divide both sides by [tex]\(c\)[/tex] to isolate [tex]\(x\)[/tex]:
[tex]\[ x = \frac{d}{c} \][/tex]
### 4. Solving [tex]\(a + 5 x = 7 x - r\)[/tex] for [tex]\(r\)[/tex]:
First, let's simplify and isolate [tex]\(r\)[/tex]:
[tex]\[ a + 5 x = 7 x - r \][/tex]
Subtract [tex]\(5 x\)[/tex] from both sides:
[tex]\[ a = 7 x - 5 x - r \][/tex]
Which simplifies to:
[tex]\[ a = 2 x - r \][/tex]
Add [tex]\(r\)[/tex] to both sides:
[tex]\[ a + r = 2 x \][/tex]
Subtract [tex]\(a\)[/tex] from both sides to isolate [tex]\(r\)[/tex]:
[tex]\[ r = 2 x - a \][/tex]
### Summary:
Combining all these results, we get:
1. [tex]\( x = 12 \)[/tex]
2. [tex]\( x = b - a \)[/tex]
3. [tex]\( x = \frac{d}{c} \)[/tex]
4. [tex]\( r = 2 x - a \)[/tex]
So, the solutions to the given equations are:
[tex]\[ (x, x \text{ in terms of } a \text{ and } b, x \text{ in terms of } c \text{ and } d, r \text{ in terms of } a \text{ and } x) = (12, b - a, \frac{d}{c}, 2x - a) \][/tex]
### 1. Solving [tex]\(\frac{3}{4} x = 9\)[/tex] for [tex]\(x\)[/tex]:
First, isolate [tex]\(x\)[/tex]:
[tex]\[ \frac{3}{4} x = 9 \][/tex]
Multiply both sides by [tex]\(\frac{4}{3}\)[/tex] to get [tex]\(x\)[/tex] alone:
[tex]\[ x = 9 \times \frac{4}{3} \][/tex]
Simplifying the right side:
[tex]\[ x = 12 \][/tex]
### 2. Solving [tex]\(a + x = b\)[/tex] for [tex]\(x\)[/tex]:
We need to express [tex]\(x\)[/tex] in terms of [tex]\(a\)[/tex] and [tex]\(b\)[/tex]:
[tex]\[ a + x = b \][/tex]
Subtract [tex]\(a\)[/tex] from both sides to isolate [tex]\(x\)[/tex]:
[tex]\[ x = b - a \][/tex]
### 3. Solving [tex]\(c x = d\)[/tex] for [tex]\(x\)[/tex]:
We need to express [tex]\(x\)[/tex] in terms of [tex]\(c\)[/tex] and [tex]\(d\)[/tex]:
[tex]\[ c x = d \][/tex]
Divide both sides by [tex]\(c\)[/tex] to isolate [tex]\(x\)[/tex]:
[tex]\[ x = \frac{d}{c} \][/tex]
### 4. Solving [tex]\(a + 5 x = 7 x - r\)[/tex] for [tex]\(r\)[/tex]:
First, let's simplify and isolate [tex]\(r\)[/tex]:
[tex]\[ a + 5 x = 7 x - r \][/tex]
Subtract [tex]\(5 x\)[/tex] from both sides:
[tex]\[ a = 7 x - 5 x - r \][/tex]
Which simplifies to:
[tex]\[ a = 2 x - r \][/tex]
Add [tex]\(r\)[/tex] to both sides:
[tex]\[ a + r = 2 x \][/tex]
Subtract [tex]\(a\)[/tex] from both sides to isolate [tex]\(r\)[/tex]:
[tex]\[ r = 2 x - a \][/tex]
### Summary:
Combining all these results, we get:
1. [tex]\( x = 12 \)[/tex]
2. [tex]\( x = b - a \)[/tex]
3. [tex]\( x = \frac{d}{c} \)[/tex]
4. [tex]\( r = 2 x - a \)[/tex]
So, the solutions to the given equations are:
[tex]\[ (x, x \text{ in terms of } a \text{ and } b, x \text{ in terms of } c \text{ and } d, r \text{ in terms of } a \text{ and } x) = (12, b - a, \frac{d}{c}, 2x - a) \][/tex]
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