Solving Linear Equations

What an equation is

You could find the missing number in 2x + 3 = x + 7 by guessing: try 1, try 2, try 3, try 4 — got it. Guessing works — this time. Now try guessing your way through 12x + 30 = 15x + 15 and lunch is over. This lesson is the machine that replaces guessing: one legal move, repeated, corners any mystery number in three or four lines. But the machine only works if every move is legal — and some very tempting moves aren't.

First, the object itself. An equation says two expressions are equal. The equals sign isn't "compute this." It's a statement that the value on the left and the value on the right are the same number.

That's a sentence: "The number 2x + 3 and the number x + 7 are the same number." Our job is to find a value of x that makes the sentence true. That value is called the solution.

Connecting expressions to equations

The last lesson simplified expressions by rewriting one expression without changing its value. Equations use the same skill, but now the expression sits on one side of an equals sign.

| Expression work | Equation work | | --------------------------------------- | ------------------------------------------------------------------------- | | Simplify 3x + 2x + 4 to 5x + 4. | Rewrite 3x + 2x + 4 = 19 as 5x + 4 = 19. | | The value stays the same for every x. | The solution stays the same because one side was simplified equivalently. |

So when you simplify a side of an equation, you are not solving yet. You are making an equivalent equation that is easier to solve.

Equivalence: the one move that's allowed

There's exactly one rule for solving:

Whatever you do to one side, do the same thing to the other. If two quantities are equal and you change them both in the same way, they're still equal. That's the equivalence principle.

Each time you apply an equivalent move, you get a new equation with the same solution as the old one. We chain those moves until the equation reads x = (a number). That number is the solution.

Careful, though: the solver below mixes real equivalent moves with tempting traps. Before each move runs, you commit — does it keep the equation equivalent, or break the balance? Every move executes either way. The original equation, in the test panel underneath, is the final judge.

Here's a worked chain on a different equation, 4x + 1 = 2x + 9. Subtract 2x from both sides:

Subtract 1 from both sides, then divide both sides by 2:

Done. x = 4 is the solution. Substitute it into the original to check: $4(4) + 1 = 17$ on the left, $2(4) + 9 = 17$ on the right. Both sides are 17, so the sentence is true.

When one side has brackets

Distribute first, then chain equivalent moves. The distributive property (covered in Simplifying Algebraic Expressions) turns

$3(x + 2)$ into 3x + 6. So to solve $3(x + 2) = x + 14$:

From there the chain is the same as before: subtract x, then subtract 6, then divide by 2. The solution is x = 4.

Run one through the solver yourself — and watch the distribute step. Both expansions on offer look plausible; only one of them is the distributive property.

Quick check before the worksheet — five true-or-false rounds on the moves this lesson lives on:

Where it shows up in real life

Splitting a campsite at Pigeon Lake. Your group books a campsite for $45 plus $8 per night. A different group books a single-night stay at a different site for $60. For how many nights would your group pay the same total as theirs?

Let n be the number of nights:

Subtract 45 from both sides:

Divide both sides by 8:

There's no whole-number of nights where the totals match. At two nights, your group has already paid more.

Renting a canoe at Wabamun. A canoe rental is $30 flat plus $12 per hour. A second outfitter charges a flat $15 plus $15 per hour. For how many hours does the second outfitter cost the same as the first? Let x be the number of hours — write the equation, then solve it here:

Worksheet

Try these to put the idea into practice. Each one ends in x = (a number). Verify by substituting back.

Going further

The same equivalence principle handles inequalities: 2x + 3 > x + 7 solves the same way, except the answer is a range of values rather than a single number. (One detail differs: multiplying or dividing both sides by a negative flips the inequality sign.)

It also handles systems of equations: two equations in two unknowns. Combining or substituting reduces the system to a single linear equation, which solves with the same chain of moves.