1 · Words into expressions
"7 less than 3 times a number" is 3n − 7. "Less than" flips the order: the 7 is taken away from something, so it comes second.
"4 times the sum of a number and 5" is 4(n + 5). "The sum" tells you to add first, so the parentheses are needed.
2 · Like terms and the distributive property
Like terms have the same variable part: 5x and −2x are like terms; 5x and 5 are not. 5x + 3 − 2x + 8 = 3x + 11.
Distribute to every term inside, including the sign: −3(2x − 4) = −6x + 12. A common slip is −6x − 12.
3 · Evaluating with negatives
Put each value in parentheses before you work it out. If a = −3, then a² − 2a = (−3)² − 2(−3) = 9 + 6 = 15.
4 · Solving equations
Undo operations in reverse order, doing the same thing to both sides. 4x − 9 = 23: add 9 to get 4x = 32, then divide by 4: x = 8.
With variables on both sides, gather them first. 7x + 2 = 3x + 18: subtract 3x to get 4x + 2 = 18, so 4x = 16 and x = 4. Check by putting 4 back in: 30 = 30.
5 · Equations from word problems
Name the unknown, then write the sentence as an equation. "A gym charges $20 to join plus $15 a month. Lee has paid $110. How many months?" 20 + 15m = 110, so 15m = 90 and m = 6.
Two people, one total: "Ana has 3 times as many cards as Bo, and together they have 64." Bo has b, Ana has 3b, so 4b = 64 and Bo has 16.
6 · Inequalities
Solve an inequality like an equation, with one extra rule: when you multiply or divide both sides by a negative number, flip the sign. −2x + 5 > 13 → −2x > 8 → x < −4.
On a number line, an open circle means the number itself is not included (< or >); a filled circle means it is (≤ or ≥).
7 · Linear patterns and the nth term
In 5, 8, 11, 14, … each term adds 3. The 20th term is the first term plus 19 steps: 5 + 19 × 3 = 62. Using 20 steps instead of 19 is the usual mistake.
A rule for the nth term: start with the step times n, then fix the start. Here 3n gives 3, 6, 9, …, which is 2 too small, so the rule is 3n + 2.
8 · Function tables
To find the rule in a table, look at how y changes when x goes up by 1. That change is the number you multiply x by. Then check one row to find what to add.
Always test the rule on every row. A rule that fits only the first pair is not the rule.
9 · Slope
Slope is the change in y divided by the change in x. Through (1, 2) and (4, 11): (11 − 2) ÷ (4 − 1) = 9 ÷ 3 = 3.
A line that rises from left to right has a positive slope; one that falls has a negative slope. Upside down (change in x over change in y) is a common trap.
10 · y = mx + b
In y = mx + b, m is the slope and b is where the line crosses the y-axis. y = −2x + 5 starts at 5 and goes down 2 for each step right.
A taxi fare of $3 plus $2 a mile is y = 2x + 3: the fixed fee is b and the rate is m.
Ready to practice?
The quiz gives you new questions each time and explains every answer. You'll also find more quizzes and full practice tests.