How can you prove that the following equation 9(x-4) - 7x = 5(3x-2) has only one solution?

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The most basic way is just to solve it. If we distribute and combine like terms on the left hand side, we get

`9(x-4)-7x=9x-9*4-7x=2x-36.` Distributing on the right hand side gives

`5(3x-2)=5*3x-5*2=15x-10,` so the whole equation simplifies to

`2x-36=15x-10.` If we subtract `2x` from both sides and add 10 to both sides, we get

`-26=13x,` and dividing both sides by 13 then gives `x=-2.`

Think about what we've really just done. We assumed that there is some number `x` that satisfies the equation and then performed basic arithmetic and algebraic operations to show that `x` can only be ```-2```, which means there is at most one solution. It's not hard to check that it is actually a solution, so we're done.

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