Year 9 · Unit 7 Equations and linear systems · Lesson 4 of 7

One, none or infinitely many

Two straight lines meet once, never or everywhere. You can tell which from the gradients and the intercepts, before you solve anything.

Number and AlgebraNo calculator

Warm-up

Three quick ones from earlier work. Try each in your head or on paper before you open the answer.

  1. Write 2x + 3y = 6 as y = mx + c.
    Answery = −(2/3)x + 2, because 3y = 6 − 2x.
  2. Are y = 4x + 1 and y = 4x − 3 parallel? How can you tell?
    AnswerYes: they have the same gradient, 4.
  3. Does (3, 7) lie on y = 2x + 1?
    AnswerYes, because 2 × 3 + 1 = 7.
Words used in this lesson
gradient
how steep a line is. For example: in y = 4x − 1 the gradient is 4.
intercept
where a line crosses the y-axis. For example: in y = 4x − 1 the intercept is −1.
parallel
lines with the same gradient that never meet. For example: y = 4x + 5 and y = 4x − 1.
infinitely many
more than any number you could count. For example: every point on a line.
coefficient
the number multiplying a letter. For example: in 6x the coefficient of x is 6.
consistent
the equations can be true at the same time. For example: x + y = 6 and x − y = 2.

Two reports that cannot both be true

A co-op sells scarves and bags, and every scarf is one price and every bag is another. Report 1 says: one scarf and two bags cost 8 hundred taka together. Report 2 says: two scarves and four bags cost 20 hundred taka together.

Predict first: can both reports be true? What would Report 2 have to say for it to agree with Report 1 exactly? Then explore with the figure: the first line is fixed, and you choose the second.

1
−1
1
one solutionthe system has
(1.8, 0.8)the solution
−2/3 and 1the two gradients

2x + 3y = 6 and x − y = 1: the gradients are −2/3 and 1, so there is one solution, at (1.8, 0.8).

Show me the steps
  1. 2x + 3y = 12 against 2x + 3y = 6: the same gradient, −2/3, and different intercepts. Parallel: no solution.
  2. 4x + 6y = 12 against 2x + 3y = 6: every term doubled, so the same line. Infinitely many solutions.
  3. x − y = 1 against 2x + 3y = 6: gradients 1 and −2/3. They cross once, at (1.8, 0.8).
  4. −2x + 3y = 6 against 2x + 3y = 6: gradients 2/3 and −2/3. They cross once, at (0, 2).

Two straight lines can meet in exactly one point, in no point, or in every point. Write both as y = mx + c and compare:

• Different gradients: one solution. The lines must cross once.
• Same gradient, different intercepts: no solution. The lines are parallel.
• Same gradient, same intercept: infinitely many solutions. The two equations describe the same line.

In standard form, check whether one equation is a multiple of the other: both sides must be multiplied by the same number for the lines to be the same.

Back to the reports: Report 2 has exactly double the scarves and bags of Report 1, so for the two to agree it would have to say 2 × 8 = 16 hundred taka. It says 20, so the two reports are parallel lines and no prices can satisfy both: somebody has made a mistake.

A common mistake: multiplying only the left side. 2x + 3y = 6 and 4x + 6y = 9 look alike, but doubling the first gives 4x + 6y = 12, not 9. Same gradient, different total: parallel, so no solution. Also, the same gradient does not always mean no solution: when the intercepts match, the lines are the same line.

Worked examples: predicting without solving

How many solutions: y = 5x + 2 and y = 5x − 7?

  1. Both gradients are 5, so the lines are parallel or the same.
  2. The intercepts are 2 and −7, which differ, so they are not the same line.
  3. No solution. Parallel lines never meet.

How many solutions: 6x + 9y = 12 and 2x + 3y = 4?

  1. Multiply the second equation by 3: 6x + 9y = 12.
  2. That is exactly the first equation, so the two lines are the same.
  3. Infinitely many solutions. Every point on the line works.

How many solutions: y = 4x − 1 and y = x + 8?

  1. The gradients are 4 and 1, which differ.
  2. Different gradients mean the lines must cross once. One solution.
  3. It is where 4x − 1 = x + 8, so 3x = 9, x = 3, y = 11.

Your go. The first step is given. How many solutions: x + 3y = 9 and 2x + 6y = 15?

  1. Double the first equation: 2x + 6y = 18.
  2. Compare that with the second equation, then decide.
Check the answer

The left sides match but the totals are 18 and 15, so the lines are parallel: no solution.

Check before you go on

1. How many solutions does the system x + y = 6 and x + y = 10 have?

Your turn

Start at Level 1 and go as far as you can. Type negative numbers with a minus sign, for example -7. A wrong answer gets a reason, not just a cross: read it, then try again.

Level 1 · Getting started

2. How many solutions does the system y = 2x + 5 and y = −x + 5 have?

3. Which pair of lines is parallel?

Level 2 · Building

4. For which value of k are the lines y = kx + 3 and y = 5x − 2 parallel?

5. Without solving, how many solutions have 2x + 4y = 10 and 3x + 6y = 15?

Level 3 · Stretch

6. 5x + 2y = 10 and 15x + 6y = c. For which value of c do they have infinitely many solutions?

7. For which value of a does 2x + 3y = 6 and ax + 6y = 9 have no solution?

Something to think about

Three lines are drawn so that every pair of them meets, but there is no single point on all three. Is that possible? Sketch one way it could look.

One way to think about it

Yes: three lines that form a triangle. Each pair meets at a corner, but no point is on all three. So a system of three equations in two unknowns can have no solution even when no two of the lines are parallel.

Exit ticket

8. A system of two linear equations has no solution. Which statement must be true?