EST I HS Math · Unit 1 · Foundational Algebra

Lesson 1.4 · Linear Graphs: Slope, Intercepts, Parallel & Perpendicular

On the paper: The most repeated topic on real EST I papers: 3–5 questions per paper

Big idea. A line is one number (its slope) and one point. Everything the EST asks — intercepts, a third point, parallel, perpendicular — is a two-line computation once you can read the slope straight off the equation.

1Slope = rise ÷ run 📈

🍼 Baby steps

1
A(2, 28), origin (0, 0)
2026 sample Q12. Two points → slope.
2
m = (28 − 0) ÷ (2 − 0) = 14
rise over run. Order does not matter as long as you subtract in the same order top and bottom.
3
through the origin: y = 14x
A line through the origin has no "+ b".
4
x = −5 → y = −70
That was the question: the y-coordinate of the point with x = −5.
5
f(2) = 1, f(−3) = 11 → m = (11 − 1)/(−3 − 2) = −2
December 2024: a "linear function with two values" is the same two-point problem in disguise.
6
y = −2x + b, 1 = −4 + b → b = 5
Plug either point in to find b. That b IS the y-intercept the question wanted.

🎛 Play with it · two points, one line

Slide A and B. Watch rise, run, slope and both intercepts update. Then switch on a parallel or perpendicular line through P and read its slope.

A x₁ -4 y₁ -1B x₂ 4 y₂ 5

slope m = rise ÷ run = 6 ÷ 8 = 0.75

y = 0.75x + 2

y-intercept (x = 0): 2 · x-intercept (y = 0): -2.67

2Read the equation without rearranging 👓

🍼 Baby steps

1
ax + by = c ⇒ slope = −a/b
Memorise this one line. 2x + 5y = 40 → slope −2/5. x − 3y − 12 = 0 → slope −1/(−3) = 1/3. No solving for y.
2
x-intercept: set y = 0 → 2x = 40 → x = 20
2026 sample Q22. An intercept is where the OTHER variable is zero.
3
y-intercept: set x = 0 → 5y = 40 → y = 8
Same idea. The paper offers 8 as a trap for the x-intercept question.
4
y − 1 = 7x through (1, a) → a − 1 = 7 → a = 8 → 2a = 16
2026 sample Q13. "Passes through" means: substitute the point. Then read the last line — they wanted 2a.

Trap · the intercept swap

Every intercept question lists both intercepts among the options. Say aloud which variable you are setting to zero before you touch the calculator.

3Parallel: same slope. Perpendicular: flip and negate. ⟂

🍼 Baby steps

1
parallel ⇒ m₂ = m₁
Parallel lines never meet because they climb at the same rate.
2
perpendicular ⇒ m₂ = −1/m₁ (m₁ · m₂ = −1)
Flip the fraction and change the sign: 3/2 → −2/3; −1/3 → 3; 2 → −1/2.
3
l: x − 3y − 12 = 0 → m = 1/3; parallel through A(0, −1): y = (1/3)x − 1
2026 sample Q17. Then test the options: (3, 0) → 0 = 1 − 1 ✓.
4
(d) 2x + ty = −1, (m) −3x − gy = 2, perpendicular → (−2/t)(−3/g) = −1 → tg = −6
October 2021. Product of slopes = −1, and the question wanted −5tg = 30. Never solve for t or g separately.
5
x + 5ny = 17 ∥ 2nx + 10y = 3 → −1/(5n) = −2n/10 → n² = 1 → n = 1
December 2024. Set the two slopes equal; n > 0 picks the root.

4Midpoint = average ⚖️

🍼 Baby steps

1
A(−20, 4), K(12, −3) → midpoint ((−20 + 12)/2, (4 − 3)/2) = (−4, 0.5)
2026 sample Q23 wanted only the x-coordinate: −4.
2
centre of a parallelogram = midpoint of a diagonal
May 2021: A(1, 2), C(6, 1) → (3.5, 1.5). The diagonals bisect each other, so either diagonal works.

5Real EST questions, solved twice 📄

Real question 1EST I HS · 2026 official sample · Q22Knowledge

What is the x-intercept of the line defined by 2x + 5y = 40?

A.2
The coefficient of x, copied.
B.8
The y-intercept (x = 0 → 5y = 40).
C.10
40 ÷ 4 — a guess.
D.20

Safe way · ~20 s

x-intercept: y = 0.
2x = 40 → x = 20.

⏱ 90-second way · ~6 s

Cover the y-term: 2x = 40 → 20.

Answer: D

Real question 2EST I HS · 2026 official sample · Q17Application

Consider the line l defined by x − 3y − 12 = 0. Let AB be a line parallel to l, where A has coordinates (0, −1). Which of the following could be the coordinates of point B?

A.(6, −1)
Gives slope 0 from A — a horizontal line, not slope 1/3.
B.(−3, 1)
Slope from A = 2/(−3) = −2/3.
C.(3, 0)
D.(−2, 1)
Slope from A = 2/(−2) = −1.

Safe way · ~45 s

Rearrange l: 3y = x − 12 → y = x/3 − 4, slope 1/3.
AB: y = x/3 − 1 (through (0, −1)).
Test: (3, 0): 0 = 1 − 1 ✓.

⏱ 90-second way · ~15 s

slope of l = −a/b = −1/(−3) = 1/3.
From A(0, −1), which option rises 1 for every 3 across? (3, 0): run 3, rise 1 ✓.

Answer: C

Real question 3EST I · December 2024 · Q6Application

Given a linear function f such that f(2) = 1 and f(−3) = 11. What is the y-intercept of f?

A.3
Averaging something — no.
B.5
C.6
11 − 5 — a slip in the substitution.
D.9
Used slope +2: 1 = 4 + b → b = −3, then dropped the sign… no consistent route.

Safe way · ~45 s

Points (2, 1) and (−3, 11). Slope = (11 − 1)/(−3 − 2) = −2.
y = −2x + b; 1 = −4 + b → b = 5.

⏱ 90-second way · ~15 s

Slope −2. From (2, 1) walk back 2 units in x to x = 0: y rises by 2 × 2 = 4 → 5.

Answer: B

Real question 4EST I · October 2021 · Section II · Q24Reasoning

Given that the two lines (d): 2x + ty = −1 and (m): −3x − gy = 2 are perpendicular, and t and g are constants, what is the value of −5tg?

A.−30
Sign lost: tg = −6 gives −5 × (−6) = +30.
B.−6
That is tg itself, not −5tg.
C.3/2
A ratio of coefficients, not the product asked.
D.30

Safe way · ~60 s

Slope of (d): −2/t. Slope of (m): −(−3)/(−g) = −3/g.
Perpendicular: (−2/t)(−3/g) = −1 → 6/(tg) = −1 → tg = −6.
−5tg = 30.

⏱ 90-second way · ~20 s

Slopes −a/b: −2/t and −3/g. Product = 6/(tg) = −1 → tg = −6 → −5tg = 30. Never find t or g.

Answer: D

Real question 5EST I · June 2021 · Section I · Q4Application

In the xy-plane, the equation of the line (d) passing through A(−1, 4) and perpendicular to the line 3x − 2y = 5 is:

A.y = (3/2)x + 11/2
Parallel slope (3/2), not perpendicular.
B.y = −(3/2)x + 5/2
Negated but not flipped.
C.y = −(2/3)x + 14/3
Right slope, wrong intercept: 4 = 2/3 + b gives b = 10/3, not 14/3.
D.y = −(2/3)x + 10/3

Safe way · ~60 s

3x − 2y = 5 → y = (3/2)x − 5/2, slope 3/2.
Perpendicular slope −2/3.
y = −(2/3)x + b through (−1, 4): 4 = 2/3 + b → b = 10/3.

⏱ 90-second way · ~20 s

slope = −a/b = −3/(−2) = 3/2 → perpendicular −2/3. Only C and D have it.
Test A(−1, 4) in D: −(2/3)(−1) + 10/3 = 2/3 + 10/3 = 4 ✓.

Answer: D

⏱ The 90-second lane

One question = 90 seconds. These are the moves that finish this topic in one line.

1ax + by = c has slope −a/b. Read it; never rearrange.
2x-intercept: cover the y-term and solve. y-intercept: cover the x-term.
3"Passes through (p, q)" = substitute p and q. Then read what they actually asked (2a, not a).
4Two points: slope = Δy/Δx. Then walk from a known point to x = 0 for the y-intercept.
5Parallel ⇒ same slope. Perpendicular ⇒ flip and negate; product −1. With unknown coefficients, set up the product — do not solve for each letter.
6Midpoint = average of x, average of y. Centre of a parallelogram = midpoint of either diagonal.
7Line through the origin: y = mx, no constant. Test options by plugging in.
8When options are equations, test the given point in each — the wrong ones fail in seconds.

Take with you

▸Slope is the rate: rise ÷ run, or −a/b straight from ax + by = c.
▸Intercepts: zero the other variable. Points on the line: substitute.
▸Parallel keeps the slope; perpendicular flips and negates it.
▸Midpoints are averages; parallelogram centres are diagonal midpoints.

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