We start with some unknown number x.
We then multiply it with 'a' to get ax
Then add on b to get ax+b
The order of this is important because we follow those operations in reverse to undo them. This will isolate x.
How do we undo the "add b" part? By subtracting b from both sides.
Undoing the "multiply by a" part means we divide both sides by 'a'.
These are called inverse operations.
Here are the steps to isolate x.
Subtracted b from both sides
Divide both sides by 'a' (we must require that to avoid division by zero errors)
And we're done. The x is all by itself now.
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Here's a slightly more concrete example
Say we had 2x+5 = 37 as our equation.
I replaced 'a' with 2, replaced b with 5, and c with 37.
Pick any three numbers you want. As long as 'a' is nonzero.
We'll subtract 5 from both sides to undo the "plus 5".
Then divide both sides by 2 to undo the "multiply by 2".
Follow PEMDAS in reverse to isolate the variable.
Subtract 5 from both sides
Divided both sides by 2
I'll leave the right hand side unevaluated so you can see how it fits the format
Three cases may happen:
(a) no solutions;
(b) infinitely many solutions;
(c) one and only one solution.
Case (a) happens when a= 0, b =/= c.
Case (b) happens when a= 0, b = c.
Case (c) happens when a =/= 0, while b and c may have arbitrary values.
In this case, x = is the unique solution.
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