https://file.aiquickdraw.com/mathbot/file/1724167315955ynrfmpn5.jpg

X-8÷x-10=x-5÷x-7$$ \int\int\,\mathrm{d}x\, $$

The equation in the image is:

\[
\frac{x - 8}{x - 10} + \frac{x - 4}{x - 6} = \frac{x - 5}{x - 9} + \frac{x - 7}{x - 4}
\]

To solve this equation, let's first attempt to simplify and cross-multiply where necessary.

### Step 1: Simplify the equation
Let's rewrite the equation to combine like terms and cross-multiply.

\[
\frac{x - 8}{x - 10} + \frac{x - 4}{x - 6} = \frac{x - 5}{x - 9} + \frac{x - 7}{x - 4}
\]

### Step 2: Find a common denominator
Since we are dealing with fractions, find a common denominator for the terms on both sides.

- The common denominator on the left side would be $(x-10)(x-6)$.
- The common denominator on the right side would be $(x-9)(x-4)$.

Let's rewrite the equation with these common denominators:

\[
\frac{(x-8)(x-6) + (x-4)(x-10)}{(x-10)(x-6)} = \frac{(x-5)(x-4) + (x-7)(x-9)}{(x-9)(x-4)}
\]

### Step 3: Simplify the numerators
Let's expand and simplify the numerators:

For the left side:

\[
(x-8)(x-6) + (x-4)(x-10) = (x^2 - 6x - 8x + 48) + (x^2 - 10x - 4x + 40)
\]

\[
= x^2 - 14x + 48 + x^2 - 14x + 40 = 2x^2 - 28x + 88
\]

For the right side:

\[
(x-5)(x-4) + (x-7)(x-9) = (x^2 - 4x - 5x + 20) + (x^2 - 9x - 7x + 63)
\]

\[
= x^2 - 9x + 20 + x^2 - 16x + 63 = 2x^2 - 25x + 83
\]

### Step 4: Compare the simplified numerators

Now the equation is:

\[
\frac{2x^2 - 28x + 88}{(x-10)(x-6)} = \frac{2x^2 - 25x + 83}{(x-9)(x-4)}
\]

Since the denominators are different and we want equality, equate the numerators:

\[
2x^2 - 28x + 88 = 2x^2 - 25x + 83
\]

### Step 5: Simplify further

Subtract $2x^2$ from both sides:

\[
-28x + 88 = -25x + 83
\]

Now, move all terms involving $x$ to one side and constants to the other:

\[
-28x + 25x = 83 - 88
\]

Simplify:

\[
-3x = -5
\]

### Step 6: Solve for $x$

\[
x = \frac{5}{3}
\]

So, the solution to the equation is:

\[
x = \frac{5}{3}
\]

---

Do you want more details or have any questions? Here are some related questions:

1. How do you cross-multiply fractions in an equation?
2. Why is it important to find a common denominator when dealing with fractions in equations?
3. What are the steps to expand binomials in polynomial equations?
4. How can you simplify polynomial expressions by combining like terms?
5. What are potential pitfalls when solving rational equations?

**Tip:** Always check the final solution by plugging it back into the original equation to ensure that both sides are equal.

Learn how to solve the equation (x - 8)/(x - 10) = (x - 5)/(x - 7) using cross-multiplication and algebraic manipulation. This problem involves fractions and polynomial expressions, suitable for high school students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation