Solving Linear Equations

$3x + 5 = 17$ is a riddle: what number, times 3 plus 5, lands exactly on 17? Guessing can stumble onto it, but mathematicians have a method — treat the equation as a balance scale and corner $x$ calmly.

The balance rule: move both sides

The equals sign is the pivot: both pans must weigh the same. You may do anything to a balance, as long as you do it to both sides:

add 5 to both sides or subtract 5 from both sides — still balanced;
multiply both sides by 3 or divide both by 3 — still balanced.

Solving has one goal: leave $x$ standing alone on one side. The tool is inverse operations — addition undoes subtraction, division undoes multiplication.

Isolating x, step by step

First, $3x + 5 = 17$ :

Subtract 5 from both sides to clear the unrelated constant: $3x = 12$ ;
Divide both sides by 3 to unhook the coefficient: $x = 4$ .

Now one with $x$ on both sides: $5x - 2 = 3x + 8$ .

Subtract $3x$ from both sides to gather the $x$ terms: $2x - 2 = 8$ ;
Add 2 to both sides: $2x = 10$ ;
Divide both sides by 2: $x = 5$ .

Motto: gather the $x$ terms, clear the constant, divide by the coefficient. Other orders sometimes work, but this one takes the fewest detours.

Checking answers, and counting solutions

Do not rush off after solving — substitute back into the original equation. For $x = 4$ : the left side gives $3 \times 4 + 5 = 17$ , and the right side is 17 too. For $x = 5$ : left $5 \times 5 - 2 = 23$ , right $3 \times 5 + 8 = 23$ . Checking is exactly Substitution in action: a solution is the number that makes the equation true.

How many solutions?

A linear equation usually has exactly one solution, but there are exceptions. $x + 1 = x + 2$ leaves $1 = 2$ after subtracting $x$ — a contradiction, so no solution. And $x + x = 2x$ holds for every $x$ — every number is a solution. On the scale: some balances can never level; others level no matter what you put on.

Equations as lines

Equations can be drawn, too. Solving $2x + 1 = 5$ asks: where does the line $y = 2x + 1$ reach height 5? The line starts at $b = 1$ and climbs with slope $m = 2$ ; where it meets height 5, the horizontal coordinate is exactly $x = 2$ . The solution of an equation is where its line meets that height line. What $m$ and $b$ mean is the theme of Equation of a Straight Line.

InteractiveEquations as Lines

Slope m1Intercept b2

y = x + 2

y-intercept

(0, 2)

x-intercept

(-2, 0)

Set the sliders to $m = 2$ and $b = 1$ , walk along the line to height 5, and read off the horizontal coordinate — that is the solution of $2x + 1 = 5$ . Try other values of $m$ and $b$ and watch the solution move.

Check yourself

Quick quiz

1. Solve 2x+3=11. What is x?
2. Solve 5x−2=3x+8. What is x?
3. How many solutions does x+1=x+2 have?