SSC CGL Maths · Coordinate Geometry

Slope and Equation of a Line

Read a line's slope and intercepts from its equation, test whether a point lies on it, and write its equation from one point and the slope.

All Coordinate Geometry notes

Why this matters

Sixteen questions in the bank, every one EASY and every one from the 2025 shifts: more than half the chapter. Each is a single substitution or a single rearrangement, so these are the marks to take in well under 36 seconds. The options are built from the slips: the wrong sign of a slope, the reciprocal, the other intercept.

Concept 1 of 4: Slope of a line

Slope is how much the line climbs for each step to the right. A line going up to the right has a positive slope, one going down has a negative slope, a flat line has slope 0, and a vertical line has no slope at all.

Definition

Through two points: slope $=$ rise over run, the change in $y$ divided by the change in $x$ . Subtract in the same order on top and bottom.
From $y = mx + c$ : the slope is $m$ , the number in front of $x$ , and $c$ is where the line cuts the y-axis.
From $ax + by + c = 0$ : rearrange to $y = -\frac{a}{b}x - \frac{c}{b}$ , so the slope is $-\frac{a}{b}$ .
Fast route: with x and y on the same side, slope $= -\frac{\text{coefficient of } x}{\text{coefficient of } y}$ . Safe only in that form: if the equation reads $ax = by + c$ , move the terms across first.
A horizontal line $y = k$ has slope 0. A vertical line $x = k$ has no slope (the run is 0).

Slope from two points and from an equation

m = \frac{y_2 - y_1}{x_2 - x_1} \qquad y = mx + c \qquad ax + by + c = 0 \Rightarrow m = -\frac{a}{b}

mthe slope
cin y = mx + c, the y-intercept

Worked example

Find the slope of the line through

(-2, 11)

and

(3, -4)

, and check that both points lie on

3x + y = 5

, whose slope should match.

Practice this conceptself-check · 5 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 18 Sep 2025, Shift 3 · Q66Easy

Example 1 · Coordinate Geometry · Slope and Equation of a Line

Find slope of line

7x + 4y = 8

Dropping the minus sign in minus a over b

For

ax + by + c = 0

the slope is

-\frac{a}{b}

. The positive fraction

\frac{a}{b}

and the upside-down

\frac{b}{a}

are always among the options.

Subtracting in a different order on top and bottom

Rise over run needs the same point first in both differences:

\frac{y_2 - y_1}{x_2 - x_1}

. Mixing the order flips the sign of the slope.

Drill 3 more on slope of a line

Concept 2 of 4: Where a line cuts the axes

Every point on the x-axis has

y = 0

and every point on the y-axis has

x = 0

. So to find where a line crosses an axis, put the other coordinate to zero.

Definition

x-intercept: put $y = 0$ and solve for $x$ . The line meets the x-axis at $(x, 0)$ .
y-intercept: put $x = 0$ and solve for $y$ . The line meets the y-axis at $(0, y)$ .
Intercept form: a line with intercepts $p$ on the x-axis and $q$ on the y-axis is $\frac{x}{p} + \frac{y}{q} = 1$ . Divide $ax + by = k$ through by $k$ to read both intercepts at once.
The intercepts and the origin make a right triangle of area $\frac{1}{2}|pq|$ .

Intercept form of a line

\frac{x}{p} + \frac{y}{q} = 1

pthe x-intercept
qthe y-intercept

Worked example

Find both intercepts of the line

3x - 7y = 21

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 14 Sep 2025, Shift 1 · Q73Easy

Example 2 · Coordinate Geometry · Slope and Equation of a Line

Find y-intercept of

4x + 5y = 20

Putting the wrong coordinate to zero

The x-intercept comes from

y = 0

, not

x = 0

. Putting

x = 0

gives the y-intercept, which the options usually print beside the right answer.

Drill 2 more on where a line cuts the axes

Concept 3 of 4: Testing a point on a line, and finding an unknown

A point lies on a line exactly when its coordinates make the equation true. So a point on the line is an equation waiting to be used: put it in, and any unknown in the line drops out.

Definition

Test a point: put its x into the equation and see whether you get its y. Both sides must match exactly.
Find an unknown: if a line with an unknown slope or intercept passes through a given point, put the point in and solve the one equation.
Fast route for 'which point lies on it': test each option in turn. When one of the options is 'all of the above', test every point before answering, not just the first that works.

Worked example

The line

y = mx - 4

passes through

(3, 17)

. Find

m

, and then decide whether

(-1, -11)

lies on the line.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 21 Sep 2025, Shift 1 · Q70Easy

Example 3 · Coordinate Geometry · Slope and Equation of a Line

y = mx + 1

passes through

(2,5)

, find m.

Stopping at the first option that fits

When 'all of the above' is an option, a point that fits does not end the question. Test every listed point; if they all fit, the answer is the 'all' option.

Swapping the coordinates when substituting

(3, 17)

the 3 is x and the 17 is y. Putting 17 in for x gives a different, wrong value of the unknown.

Drill 5 more on testing a point on a line, and finding an unknown

Concept 4 of 4: Equation of a line from a point and its slope

A slope and one point pin a line down completely: start at the point and walk with that slope. The point-slope form writes exactly that walk as an equation.

Definition

Point-slope form: the line through $(x_1, y_1)$ with slope $m$ is $y - y_1 = m(x - x_1)$ . Open the bracket to get $y = mx + c$ .
If the given point is on the y-axis, $(0, c)$ , the line is $y = mx + c$ straight away.
Through two points: find the slope first by rise over run, then use either point in the point-slope form.
Fast route with options: when all four options have the right slope, only the intercept differs. Put the given point into each and keep the one it satisfies.

Point-slope form

y - y_1 = m(x - x_1)

Worked example

Find the equation of the line through

(4, -1)

with slope 2.

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

SSC CGL · 2025 · Tier I, 19 Sep 2025, Shift 2 · Q66Easy

Example 4 · Coordinate Geometry · Slope and Equation of a Line

What is the quation of line through (-1,-2) with slope -3.

Keeping the wrong sign inside the bracket

For the point

(-4, -6)

the form is

y + 6 = m(x + 4)

: subtracting a negative coordinate turns into adding. Writing

y - 6

x - 4

gives a line through a different point.

Drill 2 more on equation of a line from a point and its slope

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (3)

Slope of a line
Slope from two points and from an equation
$m = \frac{y_2 - y_1}{x_2 - x_1} \qquad y = mx + c \qquad ax + by + c = 0 \Rightarrow m = -\frac{a}{b}$
Where a line cuts the axes
Intercept form of a line
$\frac{x}{p} + \frac{y}{q} = 1$
Equation of a line from a point and its slope
Point-slope form
$y - y_1 = m(x - x_1)$

Watch out for (6)

Dropping the minus sign in minus a over b
→ Slope of a line
Subtracting in a different order on top and bottom
→ Slope of a line
Putting the wrong coordinate to zero
→ Where a line cuts the axes
Stopping at the first option that fits
→ Testing a point on a line, and finding an unknown
Swapping the coordinates when substituting
→ Testing a point on a line, and finding an unknown
Keeping the wrong sign inside the bracket
→ Equation of a line from a point and its slope

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