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Hong KongSlope and linear relationships
Slope is the change in y divided by the change in x, and in a real-world problem it is the rate. The rule asks you to explain slope and the y-intercept, and to compare the slopes of two linear functions given algebraically and graphically.
Skill 5 asks you to explain slope and the y-intercept. Skill 7 asks you to compare two slopes across different representations. Between them, the straight line is the most heavily specified single idea in the whole Mathematics competency set.
The two forms you need
| Form | Looks like | Tells you |
|---|---|---|
| Slope-intercept | y equals mx plus b | m is the slope, b is where the line crosses the y-axis |
| Slope from two points | change in y over change in x | The rate between two given points |
For two points, subtract in the same order on top and bottom. Take the points (2, 3) and (6, 11): the change in y is 8 and the change in x is 4, so the slope is 2.
Subtracting in opposite orders gives the negative of the right answer, and that wrong answer will be one of the options. Pick a direction and stay in it.
What slope means in a word problem
This is what skill 5 is really asking. In a real-world linear model, the slope is the rate and the y-intercept is the starting amount.
- Cost of a workshop: slope is the price per participant, intercept is the flat fee
- Distance over time: slope is the speed, intercept is where you started
- Water in a tank: slope is the fill or drain rate, intercept is the level at the start
A negative slope means the quantity is falling. A slope of zero means it is not changing, which is a horizontal line, and an item can use that as an option to catch a candidate who assumes every line rises.
A tutoring service charges by the hour. The cost c in dollars for h hours is given by c equals 20h plus 45. What does the 45 represent?
- The hourly rate
- A one-time registration charge
- The total cost of 45 hours
- The number of hours included
Comparing two slopes
Skill 7 says "algebraically and graphically", so an item may give you one line as an equation and the other as a graph. The comparison is the point.
- Get both slopes into the same form, usually as numbers
- From a graph, count the rise and the run between two clean grid points
- Compare magnitude and sign separately
Step three matters. A slope of negative 5 is steeper than a slope of 2, and it is also smaller. A question asking which line is steeper wants magnitude; a question asking which slope is greater wants sign as well, and those give opposite answers.
Parallel and perpendicular
Parallel lines have equal slopes. Perpendicular lines have slopes that multiply to negative 1, which means one is the negative reciprocal of the other.
Neither appears as a named skill, so we would treat them as worth knowing rather than worth drilling. The named skills are about explaining and comparing, not about constructing perpendiculars.
If you failed the Mathematics subtest
This is the highest-yield place to restart. Five of the seven algebra skills touch linear relationships, and algebra is about 12 of the 35 questions on our split, which is our arithmetic on the rule's skill counts rather than a published weighting.
The published outcome data is reasonable here. Mathematics passed 71% of first-attempt candidates in 2025 and 83% at best attempt. A retake is a fresh single-subtest registration at USD 32.50, at least thirty days after your previous administration of that subtest, and every subtest you have already passed stays passed.
One honest warning. If the last time you saw y equals mx plus b was more than a decade ago, an evening will not do it. Slope is a concept rather than a procedure, and it needs a few sessions with a graph in front of you.
Common questions
How do I find the slope from two points?
Divide the change in y by the change in x, subtracting in the same order on both. From (2, 3) to (6, 11), the change in y is 8 and the change in x is 4, so the slope is 2. Reversing one subtraction gives the negative of the right answer.
What does the y-intercept mean in a word problem?
The starting amount, before the variable does anything. In a cost model it is the flat fee; in a distance model it is the starting position; in a tank problem it is the level at time zero. The slope is the rate that changes it.
Which line is steeper if one slope is negative?
Steepness is about magnitude, so a slope of negative 5 is steeper than a slope of 2. But negative 5 is the smaller number, so a question asking which slope is greater has the opposite answer to one asking which line is steeper.
Are parallel and perpendicular slopes tested?
Not as named skills. Parallel lines have equal slopes and perpendicular slopes multiply to negative one, both worth knowing, but the rule’s skills are about explaining slope and intercept and comparing two linear functions rather than constructing perpendiculars.