Help me understand how to solve multi-step equations including fraction variables on both sides like ex: 3/4 + 4= 2/3+11

To solve a multi-step equation with fractions on both sides, like \( \frac{3}{4} + 4 = \frac{2}{3} + 11 \), you'll need to follow a systematic approach. Here’s a step-by-step guide:

### Step 1: Eliminate the fractions (optional but makes it easier)
To eliminate the fractions, you can multiply every term by the least common denominator (LCD) of all the fractions involved. In this case, the fractions are \( \frac{3}{4} \) and \( \frac{2}{3} \).

- The denominators are 4 and 3.
- The LCD of 4 and 3 is 12.

Multiply every term in the equation by 12:

\[
12 \times \left( \frac{3}{4} \right) + 12 \times 4 = 12 \times \left( \frac{2}{3} \right) + 12 \times 11
\]

### Step 2: Simplify the equation
Now simplify each term:

- \( 12 \times \frac{3}{4} = 9 \)
- \( 12 \times 4 = 48 \)
- \( 12 \times \frac{2}{3} = 8 \)
- \( 12 \times 11 = 132 \)

So, the equation simplifies to:

\[
9 + 48 = 8 + 132
\]

### Step 3: Combine like terms
Next, combine the constants on each side of the equation:

- Left side: \( 9 + 48 = 57 \)
- Right side: \( 8 + 132 = 140 \)

So, the equation now becomes:

\[
57 = 140
\]

### Step 4: Solve for the variable
This is where you'd normally solve for the variable, but in this case, notice that \( 57 = 140 \) is not true. This indicates that the original equation has no solution.

### Conclusion:
When you encounter a contradiction like this, it means that there’s no value of the variable that will satisfy the original equation.

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### Do you want further details or have any questions?

Here are 5 additional questions you might explore:
1. What happens when both sides of the equation are equal after simplification?
2. How would you solve a similar equation if it involved variables on both sides?
3. What are the steps to solve an equation with fractions but no contradiction?
4. How can you use the distributive property in multi-step equations?
5. What are some strategies for checking your solution after solving the equation?

**Tip:** Always check your final solution by substituting it back into the original equation to ensure it works!

Help me understand how to solve multi-step equations including fraction variables on both sides like ex: 3/4 + 4 = 2/3 + 11

Learn to solve multi-step equations with fractions on both sides, such as 3/4 + 4 = 2/3 + 11. This guide provides a step-by-step approach, including elimination of fractions and combination of like terms, suitable for students in Grades 7-9.

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