Linear Equations — SPI Mathematics Formulas and Traps

Solving Linear Equations

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

Common traps

Not changing the sign when a term moves

5x - 2 = 3x + 6

gives

2x = 8

, so

x = 4

. Moving

-2

across without changing its sign gives

2x = 4

and a wrong

x = 2

.

The minus sign in front of a fraction

-\frac{x - 1}{2}

multiplied by 2 is

-(x - 1) = -x + 1

, not

-x - 1

. Bracket the numerator before you multiply.

Multiplying only some of the terms

When clearing fractions, every term is multiplied, including a number standing alone.

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

times 3 is

x + 6 = 15

, so

x = 9

.

Forgetting to flip

-2x > 6

gives

x < -3

. Dividing by

-2

without turning the sign gives

x > -3

, which fails: at

x = 0

,

-2x = 0

is not more than 6.

Flipping when the number is positive

Multiplying by 15 to clear the denominators 3 and 5 keeps the sign as it is, because 15 is positive. Only a negative multiplier flips it.

Checking in only one equation

A pair that fits one equation can miss the other.

(5, 2)

fits

x - y = 3

but not

x + 2y = 11

, since

5 + 4 = 9

.

Subtracting when the signs are opposite

If one equation has

+6y

and the other

-6y

, add the equations to remove

y

. Subtracting them doubles the y-term instead.

Graphs and Consistency of a Pair

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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}

The graph of a linear equation

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 lines

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.

Common traps

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.

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.

Keeping both roots of a square

A condition such as

p^2 = 25

gives

p = 5

or

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

In

3x - 5y = 7

the coefficient of

y

is

-5

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

Word Problems on Linear Equations

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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})

Common traps

Stopping at x

If x is the smallest piece and the question asks for the longest, work out the longest piece's expression. The value of x itself is often printed as a wrong option.

Reversing 'more than'

'A has 5 more than B' is

A = B + 5

, not

B = A + 5

. Decide who has more before you write the equation.

Mixing metres and centimetres

2.5 m is 250 cm. Adding 30 cm to a height still written as 2.5 gives nonsense, so change units first.

Reading 'half that of' backwards

'P's age is half that of Q' means

P = \frac{Q}{2}

, so Q is the older. Writing

Q = \frac{P}{2}

swaps the two people.

Using one equation for two facts

The number of coins and their value are two different facts. One equation adds the coins; the other adds what each coin is worth.

Adding to the top only

'1 is added to both the numerator and the denominator' changes

\frac{x}{y}

to

\frac{x + 1}{y + 1}

. Adding to the numerator alone answers a different question.

Linear Equations formulas

Solving Linear Equations

Solving an equation in one variable

Solving a linear inequality

Solving a pair of equations in two variables

Common traps

Graphs and Consistency of a Pair

Finding an unknown coefficient for a chosen case

The graph of a linear equation

The three cases of a pair of lines

Common traps

Word Problems on Linear Equations

One unknown from a sentence

Chains of comparisons: ages and heights

Common traps

More SPI Mathematics formula sheets