Đại số
Đã giới thiệu| English | Ý nghĩa |
|---|---|
| inequality/ɪniːˈkwɒlɪti/ | A mathematical statement comparing expressions using relations such as less than or greater than. |
| slope/sləʊp/ | hệ số góc |
| intercept/ˌɪntəˈsept/ | giao điểm |
Two taxi companies
- Taxi A charges $\$3$ plus $\$2$ per kilometre; Taxi B charges $\$6$ plus $\$1.50$ per kilometre. For which trips is Taxi B cheaper?
- Questions like this test Algebra: turning a situation into a linear equation or inequality 不等式, and reading what its numbers mean.
What this domain tests
- Linear equations and inequalities in one or two variables, systems of two linear equations, and linear functions.
- Interpreting a model: in $C = 3 + 2k$, the slope 斜率 $2$ is the cost per kilometre and the intercept 截距 $3$ is the fixed charge.
Taxi A (3 + 2x) and Taxi B (6 + 1.5x) cost the same at 6 km; B is cheaper for longer trips
Rate or starting value?
Identify the slope (rate) and intercept (starting value) in linear models from context.
A plumber charges $C = 40 + 25h$ dollars for $h$ hours of work. What does 25 represent?
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
The coefficient of $h$ is the rate: dollars per hour.
Build and solve the model
- Taxi A: $A = 3 + 2k$. Taxi B: $B = 6 + 1.5k$. B is cheaper when $6 + 1.5k < 3 + 2k$.
- Subtract $1.5k$ and $3$ from both sides: $3 < 0.5k$, so $k > 6$. Taxi B is cheaper for trips longer than 6 km.
Check the boundary: at $k = 6$, $A = 3 + 12 = 15$ and $B = 6 + 9 = 15$. They cost the same, which confirms $k = 6$ is where the two lines cross.
Solve $5x - 7 = 3x + 9$. What is $x$?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Subtract $3x$: $2x - 7 = 9$. Add 7: $2x = 16$, so $x = 8$.
Put the steps for a word problem in order.
Độ khó 3/5 · 200 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
Checking at the end catches the “wrong variable” trap.
Systems of two equations
- Solve by substitution or elimination. "Tickets cost $\$8$ (adult) and $\$5$ (child); 40 tickets made $\$260$": $a + c = 40$ and $8a + 5c = 260$.
- From the first, $c = 40 - a$. Then $8a + 5(40 - a) = 260$, so $3a = 60$, $a = 20$, $c = 20$.
A shop sold 30 notebooks, some at $\$2$ and some at $\$3$, for $\$75$ in total. How many $\$3$ notebooks were sold?
Độ khó 2/5 · 140 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
$a + b = 30$ and $2a + 3b = 75$. Then $2(30 - b) + 3b = 75$, so $b = 15$.
Traps
- Flipping the inequality: multiplying or dividing both sides by a negative number reverses the sign: $-2x > 6$ gives $x < -3$.
- Answering the wrong variable: if the question asks for the number of child tickets, do not stop at $a$.
If $-3x > 12$, then $x < -4$.
Độ khó 1/5 · 100 xu + một phần thưởng nhỏ cho lần hoàn thành đầu tiên trong ngày
True. Dividing both sides by $-3$ reverses the inequality sign: $x < -4$.
Recap
- Write the situation as an equation or inequality; know what the slope and intercept mean in context.
- Solve systems by substitution or elimination, and check which quantity the question asks for.