Question

Q23 For number 22,(Q22 What value of q will make the following equation true?
1/5(10-5q)-3=q) did you start first with the distributive property or the multiplication property of equality? Explain your selection. What would have happened if you started with the other strategy?

Answer 1

To solve the equation \( \frac{1}{5}(10 - 5q) - 3 = q \), we can choose different strategies to simplify the equation.

### Option 1: Distributive Property First

1. **Use the Distributive Property**:
We can first expand the left side:
\[
\frac{1}{5}(10) - \frac{1}{5}(5q) - 3 = q
\]
Simplifying, we get:
\[
2 - q - 3 = q
\]
Combine like terms:
\[
-q - 1 = q
\]
Now, add \(q\) to both sides:
\[
-1 = 2q
\]
Dividing by 2:
\[
q = -\frac{1}{2}
\]

### Option 2: Multiplication Property of Equality First

Alternatively, we could multiply both sides of the equation by 5 to eliminate the fraction first:
1. **Multiply Both Sides by 5**:
\[
10 - 5q - 15 = 5q
\]
This simplifies to:
\[
-5q - 5 = 5q
\]
Adding \(5q\) to both sides, we get:
\[
-5 = 10q
\]
Dividing by 10:
\[
q = -\frac{1}{2}
\]

### Conclusion

Both methods yield the same solution, \(q = -\frac{1}{2}\).

**Starting with the Distributive Property** could be advantageous if you are more comfortable dealing with expressions in their expanded form right away. However, using the **Multiplication Property of Equality** first can help avoid fractions from the outset, potentially making calculations simpler.

If we had started with the other strategy (distributive instead of multiplication or vice versa), the pathway to the solution would have been different, but the final result would remain consistent due to the properties of equality we are using. The choice of strategy doesn't affect the correctness of the solution but might influence the complexity of the steps taken to arrive at the answer.