Solve the following linear equation.

5y+105−710=−13y−1010

how many solutions

We are given the linear equation:

\[
5y + \frac{105}{710} = -13y - \frac{1010}{710}
\]

To solve this equation, we'll follow these steps:

### Step 1: Simplify the Equation

First, simplify the fractions \( \frac{105}{710} \) and \( \frac{1010}{710} \).

- Simplify \( \frac{105}{710} \):
  \[
  \frac{105}{710} = \frac{105 \div 5}{710 \div 5} = \frac{21}{142}
  \]
  
- Simplify \( \frac{1010}{710} \):
  \[
  \frac{1010}{710} = \frac{1010 \div 10}{710 \div 10} = \frac{101}{71}
  \]

Now, the equation becomes:

\[
5y + \frac{21}{142} = -13y - \frac{101}{71}
\]

### Step 2: Eliminate the Fractions

To eliminate the fractions, find the least common denominator (LCD) of 142 and 71, which is 142. Multiply the entire equation by 142:

\[
142 \times \left( 5y + \frac{21}{142} \right) = 142 \times \left( -13y - \frac{101}{71} \right)
\]

This simplifies to:

\[
142 \times 5y + 21 = 142 \times -13y - 2 \times 101
\]
\[
710y + 21 = -1846y - 202
\]

### Step 3: Solve for \( y \)

Next, combine like terms. Add \( 1846y \) to both sides:

\[
710y + 1846y + 21 = -202
\]
\[
2556y + 21 = -202
\]

Subtract 21 from both sides:

\[
2556y = -223
\]

Now, divide by 2556 to solve for \( y \):

\[
y = \frac{-223}{2556}
\]

This fraction can be simplified, but it is not a whole number.

### Conclusion

The solution to the equation is \( y = \frac{-223}{2556} \), which is a single solution. Therefore, **there is exactly 1 solution** to this equation.

Would you like more details or have any other questions?

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Here are 8 related questions:
1. How do you simplify fractions in linear equations?
2. What is the process for eliminating fractions in equations?
3. How do you solve linear equations with variables on both sides?
4. What happens if the coefficient of \( y \) on both sides of the equation is the same?
5. Can linear equations have more than one solution?
6. How do you identify if an equation has no solutions?
7. What are the steps to checking your solution to a linear equation?
8. How would the solution change if the constants on both sides of the equation were equal?

**Tip**: Always simplify fractions early in the solving process to make calculations easier!

Learn how to solve the linear equation 5y + 105/710 = -13y - 1010/710 step-by-step with detailed explanations. This problem covers solving linear equations involving fractions, suitable for students in Grades 7-9.

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