Systems with one, no or infinitely many solutions

Warm-up

Three quick questions from earlier lessons.

Discover

$y = 2x + 1$ and $y = 2x + 5$ . Can both be true at once?

No, never
Yes, always

Key idea

Two lines meet once, never, or everywhere.

Write both equations as

y = mx + b

. Different slopes: one solution. Same slope, different intercepts: parallel lines, no solution. Same slope and intercept: the same line, infinitely many solutions.

Worked example

$y = 3x - 2$ and $6x - 2y = 4$ .

y = 3x - 2

Solve the second for y.

Infinitely many solutions

Identical.

Your turn, guided

Your turn: how many solutions has $y = 4x + 1$ , $y = -x + 6$ ? Type the number.

1

Different slopes.

Practice ★

How many solutions has the system $y = 5x + 6$ and $y = 5x + 7$ ?
1. No solution
2. Exactly one
3. Infinitely many
How many solutions has the system $y = 3x + 9$ and $y = 3x + 11$ ?
1. No solution
2. Exactly one
3. Infinitely many

How many solutions has the system $y = 3x + 2$ and $y = -3x + 9$ ?
1. Exactly one
2. No solution
3. Infinitely many

Watch out!

Practice ★★

How many solutions has the system $y = 6x + 4$ and $12x - 2y = -8$ ?
1. Infinitely many
2. No solution
3. Exactly one
How many solutions has the system $y = 5x + 4$ and $10x - 2y = -8$ ?
1. Infinitely many
2. No solution
3. Exactly one

Exit ticket

Three questions to check you have got it.

How many solutions has the system $y = 5x + 6$ and $y = 5x + 7$ ?
1. No solution
2. Exactly one
3. Infinitely many
How many solutions has the system $y = 3x + 9$ and $y = 3x + 11$ ?
1. No solution
2. Exactly one
3. Infinitely many
How many solutions has the system $y = 4x + 8$ and $y = 4x + 10$ ?
1. No solution
2. Exactly one
3. Infinitely many

Summary card

How many solutions?

Different slopes: one. Parallel: none. Same line: infinitely many.

Watch out: same slope alone is not enough; check the intercepts.

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