An equation is a statement that two expressions are equal, and it is neither true nor false until the variable has a value. A solution is a value that makes it true, and substitution is the only thing that settles it. Solving is a chain of moves chosen so that the set of values passing that test never changes.
Core ideas
Everything on this course rests on one definition, and the definition is about truth rather than about procedure.
3x − 4 = 11 claims that two expressions have the same value. Until x is given a value the claim is neither true nor false — it is waiting. That is the whole difference between an equation and an expression: 3x − 4 on its own has nothing to be true about.
It is not "whatever comes out of the procedure". The procedure is a fast way of producing a candidate; substitution is what decides whether the candidate is a solution. Those are two different jobs and later lessons will produce candidates that fail.
Every legal move replaces an equation by another one with exactly the same solution set. That is the only reason the last line of the working, x = 5, says anything at all about the first.
The material
What an equation claims, what a solution is, and which moves are guaranteed to keep one.
An equation is a statement that two expressions are equal. A solution of an equation in x is a number which, written in place of every x, makes the statement true. The solution set is the set of all such numbers.
To solve an equation is to find its solution set — all of it, and nothing that does not belong in it.
An expression such as 3x − 4 names a number as soon as x is known. An equation such as 3x − 4 = 11 makes a claim about two such names. Only the second kind of thing can be true or false, and only the second kind of thing can be solved.
To test a candidate: substitute it into the original equation, evaluate the left side down to one number, evaluate the right side down to one number, and compare. Each side is worked on alone. Nothing crosses the equals sign during a check, because a check is not a solve.
The second test is as informative as the first. It establishes something definite: 4 is not a solution. "Not a solution" is a result, and most of the values you will ever test are of that kind.
0.333 and 1/3 are different numbers. The first is 333/1000. Put each into 3x = 1 and the difference shows up immediately:
So x = 0.333 is not a solution of 3x = 1, and no amount of rounding makes it one. This is why every answer on this course is written as an exact fraction. A decimal that happens to terminate, such as 0.25, is exact and perfectly acceptable; a decimal that has been cut short is a different number wearing the answer's clothes.
Two equations are equivalent when they have exactly the same solution set.
3x − 4 = 11, 3x = 15 and x = 5 are three equivalent equations. The third is the useful one only because it is equivalent to the first.
If a = b, then a + c = b + c for any number c.
If a = b, then ac = bc for any number c; and if c ≠ 0, then a/c = b/c as well.
Applied to both sides of an equation, adding or subtracting any number, and multiplying or dividing by any nonzero number, produce an equivalent equation.
The reason those moves are safe is that each one can be undone by another move of the same kind. Adding 4 is undone by subtracting 4; multiplying by 3 is undone by dividing by 3. A move that can be reversed cannot lose a solution, because the reverse move would have to put it back, and it cannot invent one either, for the same reason.
Multiplying by zero is the move that fails this test, and it fails loudly. From 2x = 6, multiplying both sides by 0 gives 0 = 0 — a statement that is true for every value of x. The original was true for one value only. Nothing can undo the multiplication, and the information is gone. That is why the theorem says c ≠ 0 and why the restriction is not decoration.
Substituting x = 5 into 3x − 4 = 11 and getting 11 = 11 proves that 5 is a solution. It proves nothing whatever about whether there are others. For this equation there are none, but that is a separate argument — it comes from the chain of equivalent equations, and “Identities and Contradictions” states it exactly.
Interactive
A value is tested, not guessed
Method
Four moves, none of which involves solving anything.
Not the third line of your working. If an earlier step was wrong, a check against that line will happily confirm a wrong answer, which is precisely the failure a check exists to catch.
Every occurrence, and put brackets round the value whenever it is negative or a fraction: 3(−2) − 4, never 3−2 − 4. The missing bracket is the commonest way a correct answer fails its own check.
Left side to a single number; right side to a single number. Fractions stay fractions. Nothing moves across the equals sign, because at this point there is nothing to rearrange.
Equal means the value is a solution of this equation. Unequal means it is not. Report the outcome either way; a failed check is a finding, not a dead end.
Nothing here is solved. Each line takes a value that has been handed to it and asks whether the statement is true.
Line 3 is the one worth sitting with. 0.333 is a good approximation to 1/3 and it is not a solution of 3x = 1, because the definition asks for a value that makes the statement true and 999/1000 = 1 is false. Approximation is a separate subject with its own rules; solving is not part of it.
Lines 1 and 4 each establish exactly one thing: that particular value passes. Whether any other value also passes is a question these checks do not touch.
For a faded check, take 4 − 3x = 10 with candidate x = −2. The substitution string is 4 − 3(−2) = 10; finish the left-side arithmetic, write the true-or-false verdict, and state exactly what that verdict does and does not establish before using the lab.
Without the lab, test both x = 3 and x = 5/2 in 4x + 1 = 13, showing the two side values and a verdict for each. Then explain why multiplying both sides by zero cannot preserve that solution set. Every later lesson produces candidates; this is how you find out what they are worth.
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