Linear equations with one, no or infinitely many solutions

Warm-up

Three quick questions from earlier lessons.

Discover

$2x + 3 = 2x + 7$ . Try a few values of $x$ . What do you notice?

It is never true
It is true for one value of $x$

Key idea

A linear equation can have one, no or infinitely many solutions.

Simplify and collect the

x

-terms. Different $x$ -coefficients: exactly one solution. Same $x$ -coefficients, different constants (you reach something like

3 = 7

): no solution. Same on both sides (you reach something like

5 = 5

): infinitely many solutions.

Worked example

$3(x - 2) = 3x - 6$ .

3x - 6 = 3x - 6

Distribute.

-6 = -6

, always true

Subtract 3x.

Infinitely many solutions

Every x works.

Your turn, guided

Your turn: how many solutions has $5x + 1 = 4x + 1$ ? Type the number.

1

Different coefficients: x = 0.

Practice ★

How many solutions has $3(x + 2) = 3x + 6$ ?
1. Infinitely many
2. Exactly one
3. No solution
How many solutions has $9(x + 2) = 9x + 18$ ?
1. Infinitely many
2. Exactly one
3. No solution

How many solutions has $2x + 7 = 2x + 13$ ?
1. No solution
2. Exactly one
3. Infinitely many

Watch out!

Practice ★★

What number $c$ makes $2x + 7 = cx + 19$ have no solution?
What number $c$ makes $3x + 6 = cx + 12$ have no solution?

Exit ticket

Three questions to check you have got it.

How many solutions has $3(x + 2) = 3x + 6$ ?
1. Infinitely many
2. Exactly one
3. No solution
How many solutions has $9(x + 2) = 9x + 18$ ?
1. Infinitely many
2. Exactly one
3. No solution
How many solutions has $6(x + 7) = 6x + 42$ ?
1. Infinitely many
2. Exactly one
3. No solution

Summary card

How many solutions?

$3 = 7$ : none. $5 = 5$ : all. $x =$ a number: one.

Watch out: a true statement like $2 = 2$ means infinitely many.

End of the lesson

Next lesson: Systems with one, no or infinitely many solutions Back to the chapter

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