SPI Mathematics · Linear Equations

Graphs and Consistency of a Pair

Every linear equation in x and y is a straight line, and comparing the ratios of two lines' coefficients tells you whether they cross once, run parallel, or are the same line.

All Linear Equations notes

Why this matters

Fourteen questions in the bank, 10 EASY and 4 MODERATE. One table of three ratios answers nearly all of them: whether two lines cross, run parallel or lie on top of each other, and so whether the pair has one solution, none, or infinitely many.

Concept 1 of 3: The graph of a linear equation

Every linear equation in x and y draws a straight line. Which letters appear tells you its direction: with both x and y it slants, with only x it stands upright, and with only y it lies flat.

Definition

$x = a$ is a vertical line, parallel to the y-axis, $a$ units from it.
$y = b$ is a horizontal line, parallel to the x-axis.
$x = 0$ is the y-axis itself, and $y = 0$ is the x-axis.
A one-letter equation is still linear: $y = 3$ is $0x + 1y = 3$ .

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.

Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 1 · Linear Equations · Graphs and Consistency of a Pair

The graph of x = -2 is a line parallel to the

Getting the axis backwards

x = 4

is parallel to the y-axis, not the x-axis. It is every point whose x is 4, and those points stand in an upright line.

Thinking a one-letter equation is not linear

y = 3

is a linear equation in two variables,

0x + 1y = 3

, and its graph is a straight line.

Drill 1 more on the graph of a linear equation

Concept 2 of 3: The three cases of a pair of lines

Two straight lines can meet once, never meet, or be the same line. Compare the three ratios of their coefficients and you know which, without drawing a graph.

Definition

For $a_1x + b_1y = c_1$ and $a_2x + b_2y = c_2$ :

Consistent means at least one solution: intersecting or coincident lines.
Inconsistent means no solution: parallel lines.
Before comparing, write both equations the same way, with the constant on the same side, and clear any fractions.

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.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 2 · Linear Equations · Graphs and Consistency of a Pair

The pair of linear equations

\frac{3x}{2} + \frac{5y}{3} = 7

and 9x + 10y = 14 is:

Comparing before clearing fractions

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

and

x + 2y = 6

look different, but doubling the first gives the second: they are the same line.

Calling coincident lines inconsistent

Coincident lines share every point, so the pair has infinitely many solutions and is consistent. Only parallel lines are inconsistent.

Drill 5 more on the three cases of a pair of lines

Concept 3 of 3: Finding an unknown coefficient for a chosen case

When a coefficient is unknown, take the case the question names, write its ratio condition, and solve for the unknown. For infinitely many solutions all three ratios must agree, so always check the third ratio too.

Definition

Unique solution: $\frac{a_1}{a_2} \ne \frac{b_1}{b_2}$ . Solve the equality and leave that value out: the answer is every value except that one.
Parallel (no solution): $\frac{a_1}{a_2} = \frac{b_1}{b_2}$ , then check that this is not equal to $\frac{c_1}{c_2}$ .
Infinitely many (coincident): all three ratios equal. A squared unknown can give two values; keep only the one that makes the third ratio match.
Dependent equations are multiples of each other: multiply one equation by any non-zero number to get the other.

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}

Worked example

Find

k

so that

kx + 4y = 8

and

3x + 6y = 12

have infinitely many solutions.

Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Linear Equations · Graphs and Consistency of a Pair

The value of 'p' for which the equations

px + 3y = p - 3, 12x + py = p

has infinitely many solutions is:

Keeping both roots of a square

A condition such as

p^2 = 25

gives

p = 5

p = -5

. Test each in the third ratio: one of them usually gives parallel lines, not the same line.

Giving the excluded value as the answer

For a unique solution the answer is every value except the one that makes the first two ratios equal. That one value gives parallel or coincident lines.

Dropping a minus sign in a coefficient

3x - 5y = 7

the coefficient of

y

-5

, not 5. Losing the sign gives the right size of answer with the wrong sign.

Drill 5 more on finding an unknown coefficient for a chosen case

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 (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

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