That's a great question! The process of solving the equation by multiplying both sides by \(4\) instead of dividing by \(\frac{1}{4}\) is indeed related. Here’s the step-by-step explanation:
When you have the equation:
\[
\frac{1}{4}(x + 3) = 5,
\]
you want to get rid of the fraction to make the equation easier to solve. Dividing by \(\frac{1}{4}\) is mathematically correct, but it can be a bit less straightforward for some people.
Why Multiply by 4?
To eliminate the \(\frac{1}{4}\), you can multiply both sides of the equation by the reciprocal of \(\frac{1}{4}\), which is \(4\).
Here's how that looks:
- Multiply Both Sides by 4:
\[
4 \times \frac{1}{4}(x + 3) = 4 \times 5.
\]
On the left-hand side, multiplying by \(4\) cancels \(\frac{1}{4}\):
\[
1(x + 3) = 20.
\]
This is because:
\[
4 \cdot \frac{1}{4} = 1.
\]
- Result:
Now you are left with:
\[
x + 3 = 20.
\]
Why Not Divide by \(\frac{1}{4}\)?
If you were to divide by \(\frac{1}{4}\), you would actually be multiplying by \(4\) (because dividing by a fraction is the same as multiplying by its reciprocal). Here’s how that looks mathematically:
\[
\frac{1}{4}(x + 3) = 5 \quad \Rightarrow \quad (x + 3) = 5 \div \frac{1}{4} = 5 \times 4 = 20.
\]
In both methods—multiplying by \(4\) or dividing by \(\frac{1}{4}\)—you arrive at the same equation, \(x + 3 = 20\). So, choosing to multiply by \(4\) is often simpler and more direct, especially when teaching or explaining the concept.
In summary, multiplying by \(4\) is just a straightforward alternative that achieves the same goal in a more intuitive and simpler way.
1 answer