Linear equations in two variables — graphical and algebraic solutions: Question bank

True or false — justify

A single linear equation in two variables has exactly one solution.

False. It has infinitely many — every point on its line. One equation gives one line; you need a second line to pin down a single crossing point.

If two lines have the same slope they can never share a point.

False. Same slope means parallel or identical. If they are the same line (all ratios equal), they share every point; only if the constant differs are they truly parallel and share none.

The equation

0 x + 0 y + 5 = 0

is a valid linear equation in two variables.

False. The definition demands

(a, b) \neq = (0, 0)

. With

a = b = 0

it reads

5 = 0

, which is never true and draws no line at all.

x = 3

is a linear equation in two variables.

True. Write it

1 \cdot x + 0 \cdot y - 3 = 0

; here

b = 0

but

a \neq = 0

, so it's valid. Its graph is a vertical line — every point with

x = 3

, any

y

If the denominator

D = a_{1} b_{2} - a_{2} b_{1} = 0

, the system definitely has no solution.

False.

D = 0

only kills the unique answer; the lines are parallel or identical. Identical lines give infinitely many solutions, so

D = 0

means "no unique solution", not "no solution".

A system with a unique solution is always "consistent".

True. Consistent means "at least one solution exists". A unique crossing point is one solution, so it qualifies — and so does the infinite-solution case.

Two equations that look completely different must represent different lines.

False.

x + 2 y = 4

and

3 x + 6 y = 12

look different but the second is just the first times 3 — same line, infinitely many shared points.

Multiplying one equation of a system by

- 1

changes its solution set.

False. Scaling both sides by any nonzero number gives an equivalent equation with the identical line, so the intersection is untouched.

Spot the error

Student says: "

3 x - y = 3

, so

y = 3 x + 3

." Where's the slip?

The sign flip is missing.

- y = 3 - 3 x \Rightarrow y = 3 x - 3

. Every term crossing the equals sign flips sign, so

+ 3

becomes

- 3

Student finds

(2, 3)

for a system, checks it in equation 1 only, and declares victory. What's wrong?

A point must satisfy both equations to be the true intersection. Checking one only proves it lies on one line; substitute into both.

Student writes the consistency test as

\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} = \frac{c _{1}}{c _{2}}

for no solution. Correct it.

That's the infinite-solutions (same line) case. No solution needs the constant ratio to differ:

\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}} \neq = \frac{c _{1}}{c _{2}}

Student computes slope of

2 x + 3 y = 12

+ \frac{2}{3}

. What went wrong?

They forgot the minus. Rearranged,

y = - \frac{2}{3} x + 4

, so slope is

- \frac{2}{3}

. From

a x + b y = c

the slope is

- \frac{a}{b}

To solve

x + 2 y = 4

and

2 x + 4 y = 12

a student adds them and marches on. Why is this doomed?

These lines are parallel (equal ratios of

a, b

but not

c

), so no point exists. Any algebra will collapse to a false statement like

0 = 4

, signalling no solution.

Student says "the elimination formula

x = \frac{c _{1} b _{2} - c _{2} b _{1}}{a _{1} b _{2} - a _{2} b _{1}}

works for every system." Push back.

It fails when

D = a_{1} b_{2} - a_{2} b_{1} = 0

— division by zero. That's exactly the parallel/same-line case where no unique

x

exists.

Why questions

Why is the graph of

a x + b y + c = 0

(with

b \neq = 0

) a straight line and not a curve?

Solving gives

y = - \frac{a}{b} x - \frac{c}{b}

, a constant slope. Equal steps in

x

produce equal steps in

y

— constant rate of change is exactly what makes a line.

Why does equal-slope-but-different-intercept mean "no solution"?

Equal slope keeps the lines forever the same steepness, so they stay a fixed gap apart; a different intercept sets that gap nonzero, so they never meet.

Why do we compare ratios

\frac{a _{1}}{a _{2}}, \frac{b _{1}}{b _{2}}

rather than the raw coefficients?

Slopes are

- \frac{a}{b}

; equal slopes mean

\frac{a _{1}}{b _{1}} = \frac{a _{2}}{b _{2}}

, which rearranges to

\frac{a _{1}}{a _{2}} = \frac{b _{1}}{b _{2}}

. The ratio captures "same direction" regardless of overall scaling.

Why is adding one equation to another a legal move in elimination?

Both sides of each equation are equal quantities; adding equal amounts to equal sides keeps equality true, so the new equation still holds for the same solution.

Why prefer substitution when a coefficient equals 1?

You can isolate that variable with no fractions (

y = \dots

cleanly), so plugging it into the other equation stays tidy and avoids arithmetic slips.

Why must

(a, b) \neq = (0, 0)

appear in the definition?

If both are zero the equation loses all

x, y

dependence — it's either always true or never true, and defines no line, so it isn't an equation "in two variables" at all.

Edge cases

What kind of system is

x = 1

and

y = 5

, and where do they meet?

A vertical line and a horizontal line — perpendicular, crossing at exactly the point

(1, 5)

: one unique solution.

Can two distinct vertical lines ever have a solution?

No.

x = 2

and

x = 5

are parallel vertical lines with no common point; there's no

x

that is both 2 and 5.

If a system reduces to

0 = 0

during elimination, what does that tell you?

The two equations were the same line. The identity

0 = 0

is always true, meaning infinitely many

(x, y)

satisfy both.

If elimination reduces to

0 = 7

, what does it mean?

A contradiction — no

(x, y)

can make it true, so the lines are parallel and the system has no solution.

(0, 0)

always a solution of

a x + b y = c

Only when

c = 0

. Plugging in gives

0 = c

, so the origin lies on the line exactly when the constant term is zero (a line through the origin).

Does swapping the order of the two equations change the answer?

No. The system is the same pair of lines; the intersection point doesn't care which equation you write first.

Connections

Slope-intercept form y = mx + c
Consistency and rank of linear systems
Simultaneous equations by matrices & determinants
Graphing lines and intercepts
Word problems — age, boat-stream, mixture
Linear inequalities in two variables