Is there a constant by which I can multiply to invert one sign?

Question

Take the equation $A(x - 7) = x + 7$. Is this possible?

In other words, is there any constant $A$ by which $(x - 7)$ can be multiplied to become $(x + 7)$?

Notes:

• $x$ is any real number.
• The original question was $x + 7 = A(x - 7) + B(x + 5)$. I thought I'd be clever and set $B$ to $0$, but I realized I had no idea what $A$ should be.

Answer 1

There is no such constant that does not depend on $x$ and $y$. Just try a few examples. For $x=3,y=1$ you would need $A=2$. For $x=5,y=1$ you would need $A=\frac 32$ The fact that these are different shows it is not possible. Why would you think it is?

Answer 2

A very quick way to see this is impossible is to consider the case that $x=y\neq 0$. Then $x-y=0$ and so $A(x-y)=0$ no matter what $A$ is, but $x+y=2x$ is nonzero.

Answer 3

To solve the actual problem consider collecting everything to one side.

You will notice that overall the equation of $(x-Ax-Bx)+(7+7A-5B)=0$ must be 0 independent of choice of $x$. What does that tell you about the relationship between $A$ and $B$? You will also note the constant term must be 0, what does that tell you about $A$ and $B$? Can you relate the expressions to solve for $A$ and $B$? (2 equations, 2 unknowns)