Which is the value for the real number x, knowing that 2x-3, 5x+1, 4x-7, are the consecutive terms of an arithmetical progression?

I looked at it this way:

Some number (let's call it y) is added to each term to get the next, right? So, let's take the first term and write and equation that states this:

2x - 3 + y = 5x + 1

Now, use inverse operations to solve for y. If you add 3 to both sides, and then subtract 2x from both sides, you're left with

y = 3x + 4

Now, to get the third term, we need to add y again, right? So, let's do some fancy substitutions.

2nd term + y = 3rd term:

5x + 1   + y  = 4x - 7

Substitute the equivalent value of the 2nd term (from the first equation) and of y that we found:

5x + 1       +       y    = 4x - 7

2x - 3 + 3x + 4  +   3x + 4  = 4x - 7

Combine like terms on the left to get:

8x  + 5 = 4x - 7

Use inverse operations to solve for x (on both sides: subtract 4x, subtract 5, and then divide by 4) and you get x = -3. Substitute this into each of the three original terms and check to see that you have an arithmetic progression (adding -5, which would be our y).

That was a more algebraic approach. I didn't do this as a first instict, but you can easily solve it with systems:

2x - 3 + y = 5x + 1

5x + 1 + y = 4x - 7

Subtract the second from the first and you eliminate the y. Your resulting equation is

-3x - 4 = x + 8

Inverse operations (on both sides: add 4, subtract x, and divide by -4) will lead to x = -3.

Always, substitute and check to ensure the accuracy of your answers!