Of course! Let's solve the equation [tex]\(\frac{x-5}{3}=\frac{x-3}{5}\)[/tex] step by step.
1. Identify the equation:
[tex]\[
\frac{x-5}{3} = \frac{x-3}{5}
\][/tex]
2. Eliminate the denominators by finding a common denominator and multiplying both sides of the equation by that common denominator.
For this equation, the common denominator of 3 and 5 is 15.
[tex]\[
15 \cdot \left( \frac{x-5}{3} \right) = 15 \cdot \left( \frac{x-3}{5} \right)
\][/tex]
3. Distribute the 15 to each side:
[tex]\[
15 \cdot \frac{x-5}{3} = 15 \cdot \frac{x-3}{5}
\][/tex]
On the left side, dividing 15 by 3 gives 5:
[tex]\[
5(x - 5)
\][/tex]
On the right side, dividing 15 by 5 gives 3:
[tex]\[
3(x - 3)
\][/tex]
So now the equation looks like this:
[tex]\[
5(x - 5) = 3(x - 3)
\][/tex]
4. Distribute the constants on both sides:
[tex]\[
5x - 25 = 3x - 9
\][/tex]
5. Move all terms involving [tex]\(x\)[/tex] to one side of the equation and constant terms to the other side:
[tex]\[
5x - 3x = -9 + 25
\][/tex]
6. Combine like terms on both sides:
[tex]\[
2x = 16
\][/tex]
7. Solve for [tex]\(x\)[/tex] by dividing both sides by 2:
[tex]\[
x = \frac{16}{2} = 8
\][/tex]
Therefore, the solution to the equation [tex]\(\frac{x-5}{3} = \frac{x-3}{5}\)[/tex] is:
[tex]\[
x = 8
\][/tex]
