Systems of Equations, SPM Additional Mathematics Form 4

A system of equations is a set of two or more equations in the same unknowns that must be satisfied together. Solving the system means finding the values that make every equation true at the same time, not one equation at a time, but all of them at once.

In SPM Additional Mathematics this is Chapter 3 of Form 4, and it turns the equation-solving you already know into a structured, multi-equation skill.

The chapter contains two content standards from the KSSM DSKP. The first, systems of linear equations in three variables, extends the familiar pair of simultaneous equations to three equations in three unknowns $x$, $y$ and $z$.

The second, simultaneous equations involving one linear equation and one non-linear equation, pairs a straight-line equation with a curve, for example a linear equation together with one containing $x^{2}$, $y^{2}$ or $xy$, and asks you to find the points where they meet.

Both parts are really about one idea: reduce a hard problem to an easier one you can already solve. For three variables you eliminate one unknown to get an ordinary pair of simultaneous equations; for the linear-and-non-linear pair you substitute the linear equation into the non-linear one to get a single quadratic.

The whole chapter is a lesson in choosing a clean route and carrying it through carefully.

Because SPM uses analytic scoring, you earn marks for every correct step of the elimination or substitution, not only for the final answer. Our teachers treat this chapter as a place to build tidy, orderly working: line up your equations, label them, and show each substitution.

A neat layout here is worth real marks, and it is a habit that pays off in every later chapter that ends in an equation to solve.

It also helps to picture what a solution means. A linear equation in three variables represents a flat surface (a plane) in space, and solving the system finds the single point where the three surfaces cross.

A linear equation paired with a non-linear one represents a straight line meeting a curve, and the solution gives the coordinates of the meeting points. Holding on to that geometric picture explains why a curve and a line can meet twice, once, or not at all, and it connects this chapter directly to coordinate geometry and quadratic functions later in the course.

Systems of Equations is organised into two content standards. These codes and titles come directly from the DSKP, recognise them so you know exactly what a question is testing.

The two standards share one engine, substitution, but apply it in different settings. Standard 3.1 is mostly careful, organised arithmetic across three equations; standard 3.2 leans on your quadratic-solving skills from Chapter 2.

Getting comfortable with both means you can attack almost any equation-solving question the paper puts in front of you.

What a three-variable system is. Three linear equations in $x$, $y$ and $z$ form a system when all three must hold together.

A solution is an ordered triple $(x, y, z)$ that satisfies every equation. In the questions you meet at this level there is normally exactly one such triple.

$$a_{1}x + b_{1}y + c_{1}z = d_{1},\quad a_{2}x + b_{2}y + c_{2}z = d_{2},\quad a_{3}x + b_{3}y + c_{3}z = d_{3}$$

Solve by elimination. The reliable method is to eliminate one variable and shrink the problem.

Combine the equations in pairs so that one unknown cancels, which leaves two equations in two unknowns, a system you already know how to solve. Solve that pair, then substitute the two values back into any original equation to recover the third unknown.

A worked idea. Take $x+y+z=6$, $2x-y+z=3$ and $x+2y-z=2$.

Adding the first and third gives $2x+3y=8$; adding the second and third gives $3x+y=5$. From the second of these $y=5-3x$; substituting gives $2x+3(5-3x)=8$, so $-7x=-7$ and $x=1$.

Then $y=2$, and the first equation gives $z=3$. The solution is $(1, 2, 3)$, and a quick check in all three original equations confirms it.

Substitution is also allowed. Instead of eliminating, you can make one variable the subject of one equation and substitute it into the other two.

Either route is fine, choose whichever keeps the arithmetic simplest, usually by starting from the equation with the neatest coefficients (a $+1$ or $-1$ is ideal).

Simultaneous linear and non-linear equations. Here you are given one linear equation and one equation that is not linear, one containing $x^{2}$, $y^{2}$ or $xy$.

The standard route is always the same: make one variable the subject of the linear equation, then substitute that expression into the non-linear equation. This leaves a single equation in one variable, which almost always rearranges into a quadratic.

$$ax^{2} + bx + c = 0$$

Solve the quadratic. Once you have $ax^{2}+bx+c=0$, solve it by factorising, by completing the square, or with the quadratic formula.

The quadratic formula is one of the results supplied on the SPM formulae page, so you do not have to memorise it, but you must know when to reach for it.

$$x = \dfrac{-b \pm \sqrt{b^{2}-4ac}}{2a}$$

Always find both variables. Each value of the first unknown gives a matching value of the second.

After solving the quadratic, substitute each root back into the linear equation to find its partner, and present your answers as pairs. For example, $x+y=6$ with $x^{2}+y^{2}=20$ gives $y=6-x$, then $x^{2}+(6-x)^{2}=20$, which simplifies to $x^{2}-6x+8=0$.

This factorises to $(x-2)(x-4)=0$, so $x=2$ or $x=4$, and the paired solutions are $(2, 4)$ and $(4, 2)$.

How many solutions to expect. Because a linear-and-non-linear pair is a line meeting a curve, it can have two solution pairs (the line cuts the curve twice), one solution pair (the line just touches the curve), or none in real numbers (the line misses the curve).

The discriminant $b^{2}-4ac$ of your reduced quadratic tells you which case you are in, a neat link back to Chapter 2.

What is given and what to memorise. Only the quadratic formula appears in the SPM formulae list; the elimination and substitution methods themselves are not formulae to be supplied, they are techniques you understand and practise until they are automatic.

Systems of Equations can be tested in either written paper. Paper 1 (3472/1) lasts 2 hours and carries 80 marks: Section A has 12 questions worth 64 marks that you answer all of, and Section B has 3 questions worth 16 marks from which you answer 2.

Paper 2 (3472/2) lasts 2 hours 30 minutes and carries 100 marks across Section A (7 questions, 50 marks, answer all), Section B (4 questions, 30 marks, answer 3) and Section C (4 questions, 20 marks, answer 2).

Across both papers the items are limited-response subjective and structured questions, they are marked using analytic scoring, and you sit them with a non-programmable scientific calculator. We do not predict how many marks any single chapter will carry, because that varies from year to year, but the substitution work in this chapter also underpins questions in coordinate geometry and quadratic functions, so the skill is worth more than its own share of marks.

This chapter suits the structured, show-your-working format especially well. Solving three equations, or reducing a pair to a quadratic and solving it, is naturally a multi-step task, and because the scoring is analytic, each correct elimination line, each clean substitution and each solved root can earn its own mark.

Set your working out in numbered lines so the examiner can follow, and always present the final answer as a complete set of values.

Most marks lost in this chapter come from a handful of avoidable habits. Read these before every practice set until they become second nature.

• Substituting the wrong way round. For a linear-and-non-linear pair, make a variable the subject of the linear equation and substitute into the non-linear one. Doing it the other way usually creates a messier equation and needless errors.
• Finding only one variable. Every root of the quadratic has a partner. After solving, substitute each value back to find the matching one and present the answers as pairs, not as a lonely list of $x$-values.
• Substituting back into the non-linear equation. To pair up values, substitute your roots into the linear equation. Substituting into the non-linear one can throw up an extra value that does not actually satisfy the original pair.
• Expanding a square carelessly. When you substitute an expression such as $6-x$ and then square it, expand fully: $(6-x)^{2}=36-12x+x^{2}$. Dropping the middle term is a very common way to lose the whole question.
• Arithmetic slips in three-variable elimination. Eliminate the same variable from both pairs of equations, keep your signs under control, and line the terms up in columns so nothing is added to the wrong place.
• Leaving out the third value or skipping the check. State all of $x$, $y$ and $z$, and substitute your solution back into every original equation. A thirty-second check catches most slips before they cost you marks.

None of these mistakes is about ability, each one is a habit, and habits are fixable. Tick them off one at a time in your practice and your accuracy climbs quickly.

Students who score well here are rarely the fastest; they are the ones who keep their working tidy and always check.

Systems of Equations rewards a steady, ordered approach. Work through these steps in order, then use the resources below to revise and test yourself.

Aim for short, frequent sessions rather than one long cram. Solve one three-variable system and one linear-and-non-linear pair each day, and the routine soon becomes automatic.

When a step feels solid, move on; when it does not, slow down and repeat it until it does. Spread over a couple of weeks, this quiet practice turns a fiddly chapter into some of your most dependable marks.

1. 1

   Warm up on two-variable pairs

   Make sure ordinary simultaneous equations by elimination and by substitution are fully fluent, the three-variable case is built directly on top of them.

2. 2

   Extend to three variables

   Practise eliminating one unknown to reduce a 3×3 system to a 2×2 one, then solve and back-substitute to recover the third value.

3. 3

   Master the linear-and-non-linear routine

   Make a variable the subject of the linear equation, substitute into the non-linear one, and reduce the pair to a single quadratic.

4. 4

   Solve the quadratic three ways

   Practise factorising, completing the square, and the quadratic formula so you can pick the fastest route for each question.

5. 5

   Pair up, check, then time yourself

   Always find both values in each pair and check in the originals, then attempt mixed practice and worked examples with a clock running.

• Systems of Equations, Revision Notes
• Systems of Equations, Common Mistakes
• Systems of Equations, Practice Questions
• Systems of Equations, Paper 2 Answering Guide
• Systems of Equations, Key Terms
• Systems of Equations, Worked Examples (Easy)
• Systems of Equations, Worked Examples (Medium)
• Systems of Equations, Worked Examples (KBAT)