Question
2/5(3x+1)=1/3-x
Answers
Answer
Could u break down steps 1 and 2
Answer
Why are we multiplying 15 then dividing by 15 I don't quite understand
Answered by
GPT-4o mini
To solve the equation \(\frac{2}{5}(3x + 1) = \frac{1}{3} - x\), we can start by eliminating the fractions by multiplying through by 15, which is a common multiple of the denominators (5 and 3).
1. Multiply the entire equation by 15:
\[
15 \cdot \frac{2}{5}(3x + 1) = 15 \cdot \left(\frac{1}{3} - x\right)
\]
2. Simplifying both sides:
\[
6(3x + 1) = 5 - 15x
\]
3. Distribute on the left-hand side:
\[
18x + 6 = 5 - 15x
\]
4. Next, we need to get all the \(x\) terms on one side and constant terms on the other side. Add \(15x\) to both sides:
\[
18x + 15x + 6 = 5
\]
\[
33x + 6 = 5
\]
5. Now, subtract 6 from both sides:
\[
33x = 5 - 6
\]
\[
33x = -1
\]
6. Finally, divide both sides by 33 to solve for \(x\):
\[
x = -\frac{1}{33}
\]
Thus, the solution to the equation is:
\[
\boxed{-\frac{1}{33}}
\]
1. Multiply the entire equation by 15:
\[
15 \cdot \frac{2}{5}(3x + 1) = 15 \cdot \left(\frac{1}{3} - x\right)
\]
2. Simplifying both sides:
\[
6(3x + 1) = 5 - 15x
\]
3. Distribute on the left-hand side:
\[
18x + 6 = 5 - 15x
\]
4. Next, we need to get all the \(x\) terms on one side and constant terms on the other side. Add \(15x\) to both sides:
\[
18x + 15x + 6 = 5
\]
\[
33x + 6 = 5
\]
5. Now, subtract 6 from both sides:
\[
33x = 5 - 6
\]
\[
33x = -1
\]
6. Finally, divide both sides by 33 to solve for \(x\):
\[
x = -\frac{1}{33}
\]
Thus, the solution to the equation is:
\[
\boxed{-\frac{1}{33}}
\]
Answered by
GPT-4o mini
Certainly! Let's break down steps 1 and 2 in detail for the equation:
\[
\frac{2}{5}(3x + 1) = \frac{1}{3} - x
\]
### Step 1: Multiplying to Eliminate Fractions
To eliminate the fractions from the equation, we find a common multiple of the denominators (5 in the first term and 3 in the second term). A convenient common multiple is 15.
We multiply every term in the equation by 15:
\[
15 \cdot \left(\frac{2}{5}(3x + 1)\right) = 15 \cdot \left(\frac{1}{3} - x\right)
\]
Now we distribute the multiplication across each term:
1. For the left side, we calculate:
\[
15 \cdot \frac{2}{5}(3x + 1)
\]
- First, simplify \(15 \cdot \frac{2}{5}\):
\[
15 \div 5 = 3 \quad \text{then} \quad 3 \cdot 2 = 6
\]
- So, this becomes:
\[
6(3x + 1)
\]
2. For the right side:
\[
15 \cdot \left(\frac{1}{3} - x\right)
\]
- We can distribute the 15:
\[
15 \cdot \frac{1}{3} - 15x
\]
- Simplifying \(15 \cdot \frac{1}{3}\):
\[
15 \div 3 = 5
\]
- Therefore, the right side becomes:
\[
5 - 15x
\]
After multiplying both sides by 15, we have:
\[
6(3x + 1) = 5 - 15x
\]
### Step 2: Simplifying the Left Side
Now, we simplify the left-hand side, \(6(3x + 1)\):
1. Distribute the 6:
\[
6 \cdot 3x + 6 \cdot 1 = 18x + 6
\]
This means our equation now looks like this:
\[
18x + 6 = 5 - 15x
\]
From here, you can follow the previous steps to combine like terms and solve for \(x\). The important parts are eliminating fractions and simplifying expressions properly. If you have any particular questions about a sub-step, feel free to ask!
\[
\frac{2}{5}(3x + 1) = \frac{1}{3} - x
\]
### Step 1: Multiplying to Eliminate Fractions
To eliminate the fractions from the equation, we find a common multiple of the denominators (5 in the first term and 3 in the second term). A convenient common multiple is 15.
We multiply every term in the equation by 15:
\[
15 \cdot \left(\frac{2}{5}(3x + 1)\right) = 15 \cdot \left(\frac{1}{3} - x\right)
\]
Now we distribute the multiplication across each term:
1. For the left side, we calculate:
\[
15 \cdot \frac{2}{5}(3x + 1)
\]
- First, simplify \(15 \cdot \frac{2}{5}\):
\[
15 \div 5 = 3 \quad \text{then} \quad 3 \cdot 2 = 6
\]
- So, this becomes:
\[
6(3x + 1)
\]
2. For the right side:
\[
15 \cdot \left(\frac{1}{3} - x\right)
\]
- We can distribute the 15:
\[
15 \cdot \frac{1}{3} - 15x
\]
- Simplifying \(15 \cdot \frac{1}{3}\):
\[
15 \div 3 = 5
\]
- Therefore, the right side becomes:
\[
5 - 15x
\]
After multiplying both sides by 15, we have:
\[
6(3x + 1) = 5 - 15x
\]
### Step 2: Simplifying the Left Side
Now, we simplify the left-hand side, \(6(3x + 1)\):
1. Distribute the 6:
\[
6 \cdot 3x + 6 \cdot 1 = 18x + 6
\]
This means our equation now looks like this:
\[
18x + 6 = 5 - 15x
\]
From here, you can follow the previous steps to combine like terms and solve for \(x\). The important parts are eliminating fractions and simplifying expressions properly. If you have any particular questions about a sub-step, feel free to ask!
Answered by
GPT-4o mini
I apologize for any confusion! Let's clarify the purpose of multiplying by 15 and why we do it step-by-step.
### The Motivation for Multiplying by 15
The goal of multiplying the entire equation by 15 is to eliminate the fractions, making the equation easier to work with. Here’s a clearer explanation of why we do this:
1. **Fractions Complicate Calculation**: Equations with fractions can be harder to solve because you have to deal with the denominators. By multiplying through by the least common multiple (LCM) of the denominators, we can simplify the equation.
2. **Maintaining Equality**: When you multiply both sides of an equation by the same non-zero number, the equality is maintained. This means that if \(A = B\), then \(15 \cdot A = 15 \cdot B\).
### Breakdown of Multiplying by 15
Let’s start from the equation:
\[
\frac{2}{5}(3x + 1) = \frac{1}{3} - x
\]
**Identify the LCM:**
The denominators in the fractions are 5 (from \(\frac{2}{5}\)) and 3 (from \(\frac{1}{3}\)). The least common multiple of 5 and 3 is 15.
**Multiply the Entire Equation by 15:**
We multiply both sides by 15 to eliminate those fractions:
\[
15 \left( \frac{2}{5}(3x + 1) \right) = 15 \left( \frac{1}{3} - x \right)
\]
Now let’s break down what this looks like:
- **Left Side**:
- \(15 \cdot \frac{2}{5}(3x + 1)\)
- Multiply \(15\) by \(\frac{2}{5}\):
- **Calculate \(15 \div 5 = 3\)**, and then multiply by \(2\):
- \(3 \cdot 2 = 6\)
- So, it simplifies to \(6(3x + 1)\).
- **Right Side**:
- \(15 \left(\frac{1}{3} - x\right)\)
- Distribute \(15\):
- First, calculate \(15 \cdot \frac{1}{3} = 5\) (as \(15 \div 3 = 5\))
- For \(-x\), multiply by \(15\): \(15 \cdot (-x) = -15x\)
- So it simplifies to \(5 - 15x\).
### After Multiplication
After performing the multiplication, we have:
\[
6(3x + 1) = 5 - 15x
\]
This eliminates the fractions, making it easier to solve.
### Why It's Not Dividing
We are not actually dividing by 15; rather, we are simply multiplying the entire equation by 15 to clear the fractions. After this step, we can solve as normal without worrying about fractions.
### Conclusion
So, to summarize, multiplying by 15 simplifies our work by eliminating fractions, making the equation straightforward. I hope this clarifies the process! If you have any more questions, don’t hesitate to ask!
### The Motivation for Multiplying by 15
The goal of multiplying the entire equation by 15 is to eliminate the fractions, making the equation easier to work with. Here’s a clearer explanation of why we do this:
1. **Fractions Complicate Calculation**: Equations with fractions can be harder to solve because you have to deal with the denominators. By multiplying through by the least common multiple (LCM) of the denominators, we can simplify the equation.
2. **Maintaining Equality**: When you multiply both sides of an equation by the same non-zero number, the equality is maintained. This means that if \(A = B\), then \(15 \cdot A = 15 \cdot B\).
### Breakdown of Multiplying by 15
Let’s start from the equation:
\[
\frac{2}{5}(3x + 1) = \frac{1}{3} - x
\]
**Identify the LCM:**
The denominators in the fractions are 5 (from \(\frac{2}{5}\)) and 3 (from \(\frac{1}{3}\)). The least common multiple of 5 and 3 is 15.
**Multiply the Entire Equation by 15:**
We multiply both sides by 15 to eliminate those fractions:
\[
15 \left( \frac{2}{5}(3x + 1) \right) = 15 \left( \frac{1}{3} - x \right)
\]
Now let’s break down what this looks like:
- **Left Side**:
- \(15 \cdot \frac{2}{5}(3x + 1)\)
- Multiply \(15\) by \(\frac{2}{5}\):
- **Calculate \(15 \div 5 = 3\)**, and then multiply by \(2\):
- \(3 \cdot 2 = 6\)
- So, it simplifies to \(6(3x + 1)\).
- **Right Side**:
- \(15 \left(\frac{1}{3} - x\right)\)
- Distribute \(15\):
- First, calculate \(15 \cdot \frac{1}{3} = 5\) (as \(15 \div 3 = 5\))
- For \(-x\), multiply by \(15\): \(15 \cdot (-x) = -15x\)
- So it simplifies to \(5 - 15x\).
### After Multiplication
After performing the multiplication, we have:
\[
6(3x + 1) = 5 - 15x
\]
This eliminates the fractions, making it easier to solve.
### Why It's Not Dividing
We are not actually dividing by 15; rather, we are simply multiplying the entire equation by 15 to clear the fractions. After this step, we can solve as normal without worrying about fractions.
### Conclusion
So, to summarize, multiplying by 15 simplifies our work by eliminating fractions, making the equation straightforward. I hope this clarifies the process! If you have any more questions, don’t hesitate to ask!
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