https://file.aiquickdraw.com/mathbot/file/1725960460672ut0msui0.jpg

Let's solve the given problems one by one:

### a. \( 2(x + 5) = 12 \)
To solve for \(x\):
1. First, divide both sides by 2:
   \[
   x + 5 = \frac{12}{2} = 6
   \]
2. Now subtract 5 from both sides:
   \[
   x = 6 - 5 = 1
   \]
So, \( x = 1 \).

### b. \( \frac{a(x + y)}{b} = C \)
To solve for \( x \):
1. Multiply both sides by \( b \):
   \[
   a(x + y) = Cb
   \]
2. Now, divide both sides by \( a \):
   \[
   x + y = \frac{Cb}{a}
   \]
3. Subtract \( y \) from both sides:
   \[
   x = \frac{Cb}{a} - y
   \]
So, \( x = \frac{Cb}{a} - y \).

---

### Practice Problems

#### 7. \( \Sigma F = ma \), solve for \( a \):
1. Divide both sides by \( m \):
   \[
   a = \frac{\Sigma F}{m}
   \]

#### 8. \( v = \frac{x}{t} \), solve for \( t \):
1. Multiply both sides by \( t \):
   \[
   vt = x
   \]
2. Now, divide both sides by \( v \):
   \[
   t = \frac{x}{v}
   \]

#### 9. \( F = \frac{mv^2}{r} \), solve for \( v \):
1. Multiply both sides by \( r \):
   \[
   Fr = mv^2
   \]
2. Divide both sides by \( m \):
   \[
   v^2 = \frac{Fr}{m}
   \]
3. Take the square root of both sides:
   \[
   v = \sqrt{\frac{Fr}{m}}
   \]

#### 10. \( F = \frac{Gm_1m_2}{r^2} \), solve for \( r \):
1. Multiply both sides by \( r^2 \):
   \[
   Fr^2 = Gm_1m_2
   \]
2. Divide both sides by \( F \):
   \[
   r^2 = \frac{Gm_1m_2}{F}
   \]
3. Take the square root of both sides:
   \[
   r = \sqrt{\frac{Gm_1m_2}{F}}
   \]

---

Would you like more details on any of these solutions, or do you have any further questions?

### Relative Questions:
1. How do you solve for a variable when it is in a denominator?
2. What steps should you follow to isolate a variable in an equation with multiple variables?
3. Can you explain how to rearrange a formula for a variable in a square or square root?
4. How do you handle equations with exponents when solving for a variable?
5. What methods can be used to check if the solution to an equation is correct?

### Tip:
Always perform the same operation on both sides of the equation to maintain the equality while solving for a variable.

Solve the given problems involving algebraic equations and physics formulas.

Learn how to solve algebraic equations and physics formulas, including ΣF = ma and F = Gm1m2 / r^2. This problem set covers essential concepts of algebra and physics, focusing on variable isolation. Suitable for Grades 9-12 students.

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