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The given linear equation is:

\[
\frac{x + 8}{10} = \frac{9}{5} + \frac{x - 3}{7}
\]

Let's solve it step by step.

### Step 1: Eliminate the denominators by finding the least common denominator (LCD).
The denominators are 10, 5, and 7. The LCD of 10, 5, and 7 is 70.

Multiply both sides of the equation by 70:

\[
70 \times \frac{x + 8}{10} = 70 \times \left( \frac{9}{5} + \frac{x - 3}{7} \right)
\]

This simplifies to:

\[
7(x + 8) = 14 \times 9 + 10(x - 3)
\]

### Step 2: Simplify both sides.
Expand both sides of the equation:

\[
7x + 56 = 126 + 10x - 30
\]

Simplify further:

\[
7x + 56 = 96 + 10x
\]

### Step 3: Solve for \(x\).
First, move all terms involving \(x\) to one side of the equation and constant terms to the other:

\[
7x - 10x = 96 - 56
\]

Simplify:

\[
-3x = 40
\]

Now, divide by -3:

\[
x = \frac{40}{-3}
\]

Thus, \(x = -\frac{40}{3}\).

### Step 4: Check the solution.
Substitute \(x = -\frac{40}{3}\) back into the original equation:

\[
\frac{\left(-\frac{40}{3}\right) + 8}{10} = \frac{9}{5} + \frac{\left(-\frac{40}{3}\right) - 3}{7}
\]

Simplifying both sides confirms that the solution holds. Therefore, the solution is correct.

Do you have any questions or would you like further details? Here are some related questions:

1. How do you find the least common denominator (LCD) in equations?
2. What are some techniques to simplify complex fractions?
3. Why is it important to check solutions after solving equations?
4. Can this method be applied to non-linear equations as well?
5. What is the difference between solving fractions and decimals in equations?

**Tip:** When dealing with fractions in equations, always aim to eliminate the denominators early by multiplying by the least common denominator (LCD).

Solve and check the linear equation: (x + 8) / 10 = 9 / 5 + (x - 3) / 7

Learn how to solve the linear equation (x + 8)/10 = 9/5 + (x - 3)/7 with a detailed step-by-step solution. This problem covers key algebraic concepts, including cross-multiplication and finding the least common denominator (LCD). Suitable for students in Grades 8-10.

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