Solving, and why the moves are legal

Lesson 2 left you with a balance and a rule: whatever you do to one pan, you do to the other, and the beam stays level. This lesson turns that into a method.

Solving an equation means finding the value that makes the claim true. 3x + 5 = 20 is true for exactly one number, and the job is to find it without guessing.

The goal, stated once

You want to end up with the letter alone on one side: x = something. That shape is an equation you can read the answer off.

Every move you make is chosen to get closer to that shape, and every move is legal for exactly one reason: it does the same thing to both pans, so the claim survives. There's no second reason. Textbooks state the same thing as properties of equality, and OpenStax's chapter 2 has them written out if you want the formal names.

Solving one, every step annotated

Solve 3x + 5 = 20.

Now the step most people skip, and the one that makes algebra safe: check it. Put 5 back into the original equation. 3(5) + 5 is 15 + 5, which is 20, and the right side is 20. The two sides are the same size, so the claim is true, so 5 is the solution.

Notice what the check does. It doesn't tell you that your method was good; it tells you that your answer is right, which is the thing you actually care about. You can get an answer by any route at all, including guessing, and the check will still confirm or destroy it.

The harder shape: a bracket, both sides, and a negative

Most real equations are messier than the first one in three ways at once: a bracket, the unknown on both sides, and a negative term. All three at once:

Solve 4(x - 2) = 2x + 6.

Check: 4(7 - 2) is 4 × 5, which is 20. And 2(7) + 6 is 14 + 6, which is 20. The sides match, so 7 is right.

Two things in that working are worth slowing down for.

Expanding the bracket. 4(x - 2) means four lots of (x - 2), so it's 4x - 8, not 4x - 2. The multiplier reaches everything inside the bracket, including the second term and its sign. Test it with a number if you ever doubt it: at x = 3, 4(3 - 2) is 4, and 4x - 8 is 12 - 8, which is 4, while 4x - 2 would be 10.

Taking 2x off both sides. You can subtract an expression as well as a number. The balance does not care what the weight is made of, only that you removed the same weight from both pans.

The minus sign belongs to the term

The most common source of arithmetic slips in solving is a sign, and it's worth meeting directly rather than being warned about.¹

Look at 7 - 3x = 1. Solve it in your head before you read the working.

It is tempting to see a 7 and a 3 and produce a 4. That's wrong, and the reason is that the minus sign is not an instruction sitting between two numbers; it belongs to the 3x. The left side is "seven, plus negative three lots of x".

So solve it properly:

Check: 7 - 3(2) is 7 - 6, which is 1. Correct.

And notice the last division. Dividing both sides by a negative is legal like every other move: it's the same operation on both pans. Minus six divided by minus three is positive two, because a negative divided by a negative is positive.

If you ever lose track of a sign, the check catches it immediately, which is the argument for checking every single time rather than only when you are unsure.

Collecting like terms, which is why simplifying exists

5x - 3 = 2x + 9 had x terms on both sides, and the first move gathered them. That gathering has a name, collecting like terms, and it exists in service of solving rather than as a topic of its own.

Like terms are terms with the same letter part. 3x and 5x are like terms and add to 8x. 3x and 5y are not, and can't be combined at all. 3x and 5x² are not either, because x and x² are different things: at x = 3, x is 3 and x² is 9.

The test, as always, is a number. Does 3x + 5x = 8x? At x = 2: the left is 6 + 10, which is 16, and the right is 16. Yes. Does 3x + 5y = 8xy? At x = 2 and y = 1: the left is 6 + 5, which is 11, and the right is 8 × 2 × 1, which is 16. No.

What people get wrong

Doing something to one side only. Not a move. The equation you end up with isn't the one you started with, so its answer is not the answer to your problem.

4(x - 2) becoming 4x - 2. The multiplier reaches every term inside the bracket.

Losing the sign on a moved term. 9 - 2x = x - 3 becomes 9 = 3x - 3, not 9 = -x - 3. If you are unsure, add the term to both sides explicitly rather than "moving it over", which is a description of the result rather than a move.

Reading 7 - 3x as a subtraction between 7 and 3. The minus belongs to the 3x.

Not checking. Every solution can be verified in ten seconds by substitution, and the check is the only step that can't be fooled by a confident wrong method.

Connections

Lesson 2's balance is the whole justification for every move here. If you ever forget why a step is allowed, put the equation back on the scales and ask what you did to each pan.

Lesson 1's habit of substituting a number is what the check is, and it's also how you settle any question about whether two expressions are the same.

Lesson 4 draws relationships as lines, and solving turns up there as finding where a line reaches a particular height.

The lesson after that writes equations from sentences, which is the hard half of this subject: solving an equation you were handed is mechanical, and producing the right equation is where the thinking is.

Sources

[1] The lesson's shape comes from John Sweller and Graham Cooper, "The use of worked examples as a substitute for problem solving in learning algebra", Cognition and Instruction 2(1), 1985. Across five experiments with study time held equal, learners who studied worked examples went on to solve similar problems in about half the time and with roughly a fifth of the errors. That is why every idea here arrives as a worked example first, then as one with the last step removed for you to supply, and only then as a problem.

Go deeper

• OpenStax, Elementary Algebra 2e, chapter 2, free online, which works through the same moves with many more examples and states them as formal properties of equality.
• Khan Academy, solving equations, for practice with immediate feedback.