Linear Equations with Letter Coefficients

What this move does

Besides the unknown $x$ , an equation may contain other letters (such as $m$ ). Such a letter stands for a number that is known but not stated. Any linear equation can then be rearranged into

a x = b

where $a$ and $b$ are expressions in that letter. Now you may not just divide; you have to split into cases first:

Rearrange: move the terms with $x$ to the left and everything else to the right, and collect them into $a x = b$ .
Ask: when is the coefficient $a$ equal to $0$ ?
When $a \neq = 0$ , $x = \frac{b}{a}$ , and that is the only solution.
When $a = 0$ , look at $b$ : if $b \neq = 0$ there is no solution; if $b = 0$ every number is a solution.

A graph makes this clearer. Draw the left side $a x$ as the slanted line $y = a x$ through the origin, and the right side $b$ as the horizontal line $y = b$ . The solution is the $x$ -coordinate of their intersection. Drag $a$ and $b$ yourself and find all three cases.

Drag

a

0

: the slanted line lies flat. Now drag

b

Predict first

Solve $a x = a$ for $x$ . Sam divides both sides by $a$ and gets $x = 1$ . Is his answer complete?

Complete: the solution is

x = 1

Not complete: when

a = 0

, every number is a solutionNot complete: when

a = 0

, there is no solution

Choose an answer first; then you can see the result.

Work through one

Example: solve m²x − 1 = x + m for x

Step 1 / 6

Rearrange into ax = b
Move the terms with $x$ to the left and the rest to the right: $m^{2} x - x = m + 1$ . Take $x$ out on the left.

$(m^{2} - 1) x = m + 1$
When is the coefficient 0?
m ≠ ±1: divide safely
m = 1: no solution
m = −1: infinitely many solutions
Write out all three cases

Which letter is the unknown?

When there are many letters, first say clearly which one is the unknown; all the others are treated as known numbers. The same equality, with a different letter in the lead role, is a different equation.

Take the distance formula $s = v t + s_{0}$ . Treat $t$ as the unknown: $v t = s - s_{0}$ , which has exactly the form $a x = b$ , with coefficient $v$ . So

v \neq = 0 gives t = \frac{s - s _{0}}{v} .

And when $v = 0$ (not moving at all)? If $s \neq = s_{0}$ you never get there: no solution. If $s = s_{0}$ you are “there” at every moment, so $t$ can be any number. The same three endings as above.

When it fails

Break it

Counterexample lab: remove one condition

Conclusion: the equation $a x = b$ has exactly one solution, $x = \frac{b}{a}$

When $a$ is not $0$ , the slanted line and the horizontal line always meet, and they meet only once. Drag $b$ : the intersection just moves.

Practice

Exercise 1Basic

The equation $(k - 3) x = 5$ in $x$ has no solution. Find $k$ .

Exercise 2Basic

The equation $(a + 2) x = 3 a + 6$ in $x$ has infinitely many solutions. Find $a$ .

Exercise 3Advanced

The equation $a (x - 1) = 2 x + 3$ in $x$ has the solution $x = 4$ . Find $a$ . (Give your answer as a fraction.)

Exercise 4Challenge

Discuss the solutions of the equation $m^{2} x + 2 = 4 x + m$ in $x$ . Write all three cases on paper, then fill in here: for which value of $m$ does the equation have no solution?

What this move does

Work through one

Example: solve m²x − 1 = x + m for x

Which letter is the unknown?

When it fails

Counterexample lab: remove one condition

Practice

Mistake Museum · Exhibits from this stop

Dividing both sides by a letter that might be 0

Saying “no solution” as soon as the coefficient is 0