SPI Mathematics · Free notes

Linear Equations: SPI Mathematics

Linear Equations has 29 questions in the SPI bank, and four of them are from the 2022 paper. None is HARD: 20 are EASY and 9 MODERATE. Fourteen test one table, the three ratios that decide whether a pair of lines crosses, runs parallel or is the same line, so that table is worth more marks here than any trick of algebra. The four questions from the 2022 paper are different: they are sentences about ages, heights, land and animals, and the marks go to whoever turns each sentence into an equation carefully.

The handout has every subtopic, worked example, formula and trap with answers shown. The formula sheet is the chapter’s formulas and traps alone, on one page.

SPI home All formulas on one page

Subtopic notes

Formula & revision sheet

6 formulas · 2 reference tables · 20 gotchas across all subtopics — the exam-eve cheat-sheet

Solving Linear Equations

Formulas (3)

Solving an equation in one variable · One unknown on each side $ax + b = c \;\Rightarrow\; x = \dfrac{c - b}{a} \qquad ax + b = cx + d \;\Rightarrow\; x = \dfrac{d - b}{a - c} \quad (a \ne c)$
Solving a linear inequality · Flipping rule $a < b \text{ and } k < 0 \;\Rightarrow\; ka > kb$
Solving a pair of equations in two variables · Solution of a pair, and the solution check $x = \dfrac{c_1b_2 - c_2b_1}{a_1b_2 - a_2b_1}, \quad y = \dfrac{a_1c_2 - a_2c_1}{a_1b_2 - a_2b_1} \qquad (x_0, y_0) \text{ solves } ax + by = c \text{ when } ax_0 + by_0 = c$

Watch out for (7)

Not changing the sign when a term moves
→ Solving an equation in one variable
The minus sign in front of a fraction
→ Solving an equation in one variable
Multiplying only some of the terms
→ Solving an equation in one variable
Forgetting to flip
→ Solving a linear inequality
Flipping when the number is positive
→ Solving a linear inequality
Checking in only one equation
→ Solving a pair of equations in two variables
Subtracting when the signs are opposite
→ Solving a pair of equations in two variables

Graphs and Consistency of a Pair

Formulas (1)

Finding an unknown coefficient for a chosen case · Conditions on the coefficients $\text{one: } \frac{a_1}{a_2} \ne \frac{b_1}{b_2} \qquad \text{none: } \frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2} \qquad \text{infinitely many: } \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

Reference tables (2)

The graph of a linear equation6 rows

Equation	Graph	Example
$ax + by = c$ , with $a$ and $b$ not zero	A slanting straight line	$2x + y = 4$
$x = a$	A vertical line, parallel to the y-axis	$x = 3$
$y = b$	A horizontal line, parallel to the x-axis	$y = 5$
$x = 0$	The y-axis itself	every point $(0, y)$
$y = 0$	The x-axis itself	every point $(x, 0)$
$y = mx$	A line through the origin	$y = 2x$

x = a

stands up and

y = b

lies flat: each line is parallel to the axis of the letter that is missing from its equation.

The three cases of a pair of lines3 rows

Ratios	Lines	Solutions	Pair is
$\frac{a_1}{a_2} \ne \frac{b_1}{b_2}$	Intersecting	Exactly one	Consistent
$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$	Coincident (the same line)	Infinitely many	Consistent (dependent)
$\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}$	Parallel	None	Inconsistent Two vertical lines such as $x = 1$ and $x = 4$ are parallel too, so a pair like that has no solution.

Compare the a and b ratios first. Only when they are equal does the c ratio decide.

Watch out for (7)

Getting the axis backwards
→ The graph of a linear equation
Thinking a one-letter equation is not linear
→ The graph of a linear equation
Comparing before clearing fractions
→ The three cases of a pair of lines
Calling coincident lines inconsistent
→ The three cases of a pair of lines
Keeping both roots of a square
→ Finding an unknown coefficient for a chosen case
Giving the excluded value as the answer
→ Finding an unknown coefficient for a chosen case
Dropping a minus sign in a coefficient
→ Finding an unknown coefficient for a chosen case

Word Problems on Linear Equations

Formulas (2)

One unknown from a sentence · Fixed part plus a rate $y = a + bx \quad (a = \text{fixed part},\ b = \text{rate per unit}) \qquad x + (x + 1) + (x + 2) = 3x + 3$
Chains of comparisons: ages and heights · Comparison sentences as equations $A = \tfrac{1}{2}B \;\Rightarrow\; B = 2A \qquad A = B + k \ (A \text{ is } k \text{ older}) \qquad s - m = t - s \;\Rightarrow\; m + t = 2s \ (s \text{ halfway})$

Watch out for (6)

Stopping at x
→ One unknown from a sentence
Reversing 'more than'
→ One unknown from a sentence
Mixing metres and centimetres
→ Chains of comparisons: ages and heights
Reading 'half that of' backwards
→ Chains of comparisons: ages and heights
Using one equation for two facts
→ Two unknowns, two equations
Adding to the top only
→ Two unknowns, two equations

Question weightage by concept

9 concepts · 29 questions — where the marks actually sit, so you know what to drill first

Solving Linear Equations8 questions · 28%

Concept	Qs	Share
Solving an equation in one variable	4	14%
Solving a pair of equations in two variables	3	10%
Solving a linear inequality	1	3%

Graphs and Consistency of a Pair14 questions · 48%

Concept	Qs	Share
The three cases of a pair of lines	6	21%
Finding an unknown coefficient for a chosen case	6	21%
The graph of a linear equation	2	7%

Word Problems on Linear Equations7 questions · 24%

Concept	Qs	Share
Two unknowns, two equations	3	10%
One unknown from a sentence	2	7%
Chains of comparisons: ages and heights	2	7%

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