Solve equations with variables on each side

To solve equations with variables on each side, follow these steps:

1. **Simplify both sides**: Distribute and combine like terms on both sides of the equation if needed.
2. **Move variables to one side**: Use addition or subtraction to bring all variable terms to one side of the equation.
3. **Isolate the variable**: Use addition, subtraction, multiplication, or division to solve for the variable.
4. **Check your solution**: Substitute the solution back into the original equation to verify.

### Example Problem:

Solve \( 3x + 5 = 7x - 9 \).

#### Step-by-step solution:
1. Simplify both sides (no need in this case, as both sides are already simplified).
2. Move variables to one side:
   Subtract \( 3x \) from both sides:
   \[
   5 = 4x - 9
   \]

3. Move constants to the other side:
   Add \( 9 \) to both sides:
   \[
   14 = 4x
   \]

4. Isolate the variable:
   Divide both sides by \( 4 \):
   \[
   x = \frac{14}{4} = \frac{7}{2}
   \]

5. Check the solution:
   Substitute \( x = \frac{7}{2} \) back into the original equation:
   \[
   3\left(\frac{7}{2}\right) + 5 = 7\left(\frac{7}{2}\right) - 9
   \]
   Simplify both sides:
   \[
   \frac{21}{2} + 5 = \frac{49}{2} - 9
   \]
   Convert to a common denominator:
   \[
   \frac{31}{2} = \frac{31}{2}
   \]
   The solution checks out: \( x = \frac{7}{2} \).

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### Related Questions:
1. How do you handle equations with fractions or decimals?
2. What if there are parentheses in the equation?
3. How do you deal with equations that have no solution?
4. Can equations have infinite solutions? How do you identify this?
5. How do you set up equations from word problems involving variables on both sides?

**Tip:** Always check for extraneous solutions, especially when solving equations that involve squaring or square roots.

Master solving equations with variables on both sides. This comprehensive guide includes step-by-step instructions, examples, and verification techniques for Grades 7-9 students.

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