We are given the equation:

$3(x - 5) = 5(x + 7)$

Step 1: Expand both sides of the equation

First, distribute the constants 3 and 5 over the parentheses.

$3(x - 5) = 3x - 15$ $5(x + 7) = 5x + 35$

So, the equation becomes:

$3x - 15 = 5x + 35$

Step 2: Move all terms involving $x$ to one side

Subtract $3x$ from both sides to get the $x$-terms on one side.

$-15 = 5x - 3x + 35$ $-15 = 2x + 35$

Step 3: Move constant terms to the other side

Now, subtract 35 from both sides to isolate the term with $x$.

$-15 - 35 = 2x$ $-50 = 2x$

Step 4: Solve for $x$

Divide both sides by 2 to find $x$.

$x = \frac{-50}{2} = -25$

So, the solution is:

$x = -25$

Step 5: Check the solution

Substitute $x = -25$ back into the original equation:

$3(-25 - 5) = 5(-25 + 7)$ $3(-30) = 5(-18)$ $-90 = -90$

Since both sides are equal, the solution is correct.

Final Answer:

$x = -25$

Would you like further details on any of the steps, or do you have any questions?

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5 Related Questions:

1. How do you solve quadratic equations in a similar manner?
2. What is the method for solving absolute value equations?
3. Can you explain how to graph a linear equation like this one?
4. What happens if we had more terms on one side of the equation?
5. How do you handle equations with variables on both sides that have coefficients greater than 1?

Tip: Always check your solution by substituting it back into the original equation to ensure that both sides are equal!