For which value of x is 0.055x - 570 equal to (x - 78000)*0.055 + 1280?

Step-by-step solution

1. 1

   Expand the right-hand side into slope-intercept form. Distribute $0.055$ over the bracket:

   $(x-78000)\cdot 0.055+1280=0.055x-4290+1280=0.055x-3010$

   (the product $78000\times 0.055=4290$ is exact).

2. 2

   Set the two simplified expressions equal.

   $0.055x-570=0.055x-3010$

3. 3

   Cancel the $x$-terms and read the result. Subtracting $0.055x$ from both sides leaves

   $-570=-3010$

   a statement with no $x$ in it that is simply false. When the variable vanishes and leaves a false numeric statement, the equation has no solution — as opposed to vanishing into a true statement such as $0=0$, which would mean every $x$ works.

4. 4

   Explain it structurally. Both sides are lines with the same slope $0.055$ but different intercepts $-570$ and $-3010$. Parallel distinct lines never intersect, so there is no crossing point to find.

5. 5

   Quantify the permanent gap. The difference is a constant, independent of $x$:

   $\left(0.055x-570\right)-\left(0.055x-3010\right)=2440$

   Checked at $x=0$: $-570$ versus $-3010$, gap $2440$. At $x=78000$: $3720$ versus $1280$, gap $2440$. The first expression is permanently $2440$ larger, so equality is impossible.