Linear Inequalities in Two Variables · Form 4
Linear Inequalities in Two Variables: Paper 2 Answering Guide
How Linear Inequalities in Two Variables appears in Mathematics Paper 2 (Subjective), and how to lay out your working so you earn every method mark.
How it is examined
Paper 2 typically asks you to draw two or three inequalities and identify the region satisfying all of them, sometimes from a real constraint like a shop’s stock limits. Neat lines and a clearly labelled region earn the marks; a rushed graph loses them.
Showing your working
Mathematics Paper 2 (Subjective) is worth 100 marks and gives marks for the steps, not only the answer. Write each line clearly: state the formula or rule, substitute the numbers, then simplify.
If the question carries units, carry them through to the final line.
The question
A workshop assembles x tables and y chairs in one day. The total number of pieces assembled is at most 18.
The number of chairs is at least twice the number of tables. It also makes at least 3 tables.
(a) Write three inequalities, other than x ≥ 0 and y ≥ 0, that satisfy all the conditions. (b) Find the coordinates of the point where the lines y = 2x and x + y = 18 meet.
(c) Determine whether the point (4, 10) lies in the region R that satisfies all the inequalities.
Mark-earning layout, line by line
- (a) Total at most 18 → state the constraint: x + y ≤ 18.
- (a) Chairs at least twice tables → y ≥ 2x.
- (a) At least 3 tables → x ≥ 3. (Three inequalities: x + y ≤ 18, y ≥ 2x, x ≥ 3.)
- (b) Solve y = 2x and x + y = 18 together → substitute y = 2x: x + 2x = 18.
- (b) Simplify → 3x = 18 → x = 6; then y = 2(6) = 12.
- (b) Answer: the lines meet at (6, 12).
- (c) Test (4, 10) in each inequality → x + y = 4 + 10 = 14 ≤ 18 (true); y = 10 ≥ 2(4) = 8 (true); x = 4 ≥ 3 (true).
- (c) All three hold, so (4, 10) lies inside the region R.
Source:SPM: Format Pentaksiran mulai 2021, Matematik (1449)
Frequently asked questions
How are the method marks usually split in a region question?
Broadly, marks go to forming each inequality correctly, drawing each boundary with the right solid or dashed style, shading and identifying the correct overlap R, and any calculation such as an intersection or a test. Each stage is credited separately, so partial work still scores, never leave a part blank because the graph looks hard.
Do I have to find the intersection by algebra, or can I just read it off the graph?
If the question says 'find the coordinates', show the algebra, substitute one line into the other and solve, as in x + 2x = 18. A value merely read off the grid earns little without working, and if the point is not on a grid line you will misread it.
Read the graph only to check your algebraic answer looks right.
The question says 'hence find the maximum value' after shading, what do I do?
The maximum or minimum of an expression like 2x + 3y over region R sits at a corner (vertex) of R. Find the coordinates of each vertex where two boundaries meet, substitute every vertex into the expression, and compare the results.
The largest value is the maximum; state which vertex gives it. Always test all corners, not just one.