Question
Fill in the spaces to solve each step of the problem 1/4(x+3)=5
.(2 points)
Step 1: (_)14(x+3)=5(_).
Step 2: x+3=__
Step 3: Subtract __ from both sides.
Step 4: x= __
fill in the blanks __
.(2 points)
Step 1: (_)14(x+3)=5(_).
Step 2: x+3=__
Step 3: Subtract __ from both sides.
Step 4: x= __
fill in the blanks __
Answers
GPT-4o mini
Let's solve the equation step by step and fill in the blanks.
The original equation is:
\[
\frac{1}{4}(x+3)=5.
\]
**Step 1**: Multiply both sides by 4 to eliminate the fraction:
\[
(4)\cdot\frac{1}{4}(x+3) = 5(4).
\]
This simplifies to:
\[
1(x+3) = 20.
\]
So in this step, you can fill in the blanks as follows:
\[
(4)14(x+3)=5(4).
\]
**Step 2**: From \(1(x+3) = 20\), we can simplify to:
\[
x+3=20.
\]
**Step 3**: Subtract 3 from both sides:
\[
x + 3 - 3 = 20 - 3.
\]
So we fill in the blank:
\[
\text{Subtract } 3 \text{ from both sides}.
\]
**Step 4**: This gives us:
\[
x = 17.
\]
Putting it all together, the filled-in blanks would be:
1. (4)14(x+3)=5(4).
2. x+3=20.
3. Subtract 3 from both sides.
4. x=17.
The original equation is:
\[
\frac{1}{4}(x+3)=5.
\]
**Step 1**: Multiply both sides by 4 to eliminate the fraction:
\[
(4)\cdot\frac{1}{4}(x+3) = 5(4).
\]
This simplifies to:
\[
1(x+3) = 20.
\]
So in this step, you can fill in the blanks as follows:
\[
(4)14(x+3)=5(4).
\]
**Step 2**: From \(1(x+3) = 20\), we can simplify to:
\[
x+3=20.
\]
**Step 3**: Subtract 3 from both sides:
\[
x + 3 - 3 = 20 - 3.
\]
So we fill in the blank:
\[
\text{Subtract } 3 \text{ from both sides}.
\]
**Step 4**: This gives us:
\[
x = 17.
\]
Putting it all together, the filled-in blanks would be:
1. (4)14(x+3)=5(4).
2. x+3=20.
3. Subtract 3 from both sides.
4. x=17.
why is it in step one it isnt divide by 1/4? why is just 4? explain
GPT-4o mini
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:
1. **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:
\[
4 \cdot \frac{1}{4} = 1.
\]
2. **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.
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:
1. **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:
\[
4 \cdot \frac{1}{4} = 1.
\]
2. **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.
alr thank you!
GPT-4o mini
You're welcome! I'm glad I could help. If you have any more questions or if there's anything else you'd like to know, feel free to ask!