Chapter 5 of 8

All 8 chapters
  1. 1Coordinates, readunderstood
  2. 2All four quarters of the grid, readunderstood
  3. 3Shapes on a grid, readunderstood
  4. 4From a rule to a line, readunderstood
  5. 5Lines like x = 3 and y = −2, readunderstood
  6. 6Steepness: the gradient, readunderstood
  7. 7The equation y = mx + c, readunderstood
  8. 8Graphs from real life, readunderstood

M3.9.5 Chapter 5

Lines like x = 3 and y = −2

What does a line look like when x is always 3? And where do two lines meet?

Your guess

Pick one before you read on. It is fine to be wrong.

Your guess

  • You picked A. Read on. At the end, the chapter comes back to your guess.

  • You picked B. Read on. At the end, the chapter comes back to your guess.

  • You picked C. Read on. At the end, the chapter comes back to your guess.

By the end you can

  • Draw lines like x = 3 and y = −2
  • Find where two straight lines cross

In the last chapter the rule gave a different y for each x. Now look at a rule with only one letter: x = 3. It says that x is 3 for every point. The y number can be anything.

Think of the points (3, −2), (3, 1) and (3, 3). They all have x = 3. They sit one above the other, so they lie on an upright line. An upright line is vertical.

Draw it first

Draw the line x = 3

  1. A grid with the x axis across and the y axis up. Step 1: the points (3, −2), (3, 1) and (3, 3), all above 3 on the x axis. Step 2: the straight upright line x = 3 through them.−5−4−3−2−112345−5−4−3−2−1123450xy

    1 Pick some y numbers. Plot points that all have x = 3, such as (3, −2), (3, 1) and (3, 3).

  2. A grid with the x axis across and the y axis up. Step 1: the points (3, −2), (3, 1) and (3, 3), all above 3 on the x axis. Step 2: the straight upright line x = 3 through them.x = 3−5−4−3−2−112345−5−4−3−2−1123450xy

    2 Join them with a ruler. The line is upright and goes through 3 on the x axis. It goes on both ways.

Now take y = −2. Every point has y = −2, and x can be anything. The points (−4, −2), (0, −2) and (3, −2) lie side by side. They lie on a flat line. A flat line is horizontal.

A grid with the x axis across and the y axis up. The points (−4, −2), (0, −2) and (3, −2) and the flat line y = −2 through them, 2 below the x axis.y = −2−5−4−3−2−112345−5−4−3−2−1123450xy
All the points of y = −2 are 2 below the x axis.

Worked example

Draw the line y = −2 and say where it crosses the y axis.

  1. Every point has y = −2. Plot some, such as (−4, −2), (0, −2) and (3, −2).
  2. Join them with a ruler. The line is flat.
  3. The line crosses the y axis where x = 0. That is the point (0, −2).

Answer: (0, −2)

A way to remember. The line x = 3 stands on the 3 of the x axis, like a post. The line y = −2 lies at −2 on the y axis, like a shelf. The x axis is the line y = 0, and the y axis is the line x = 0.

Two straight lines that do not run the same way cross at one point. To find it, draw both lines and read the place where they meet. Then check that point in both rules.

Worked example

Find where the lines y = 2x − 1 and y = −x + 5 cross.

  1. Draw both lines on the same grid. Make a table of values for each one.
  2. They meet at the point (2, 3). Read it off the graph.
  3. Check it in the first rule: 2 × 2 − 1 = 3. This agrees with y = 3.
  4. Check it in the second rule: −2 + 5 = 3. This agrees with y = 3.

Answer: (2, 3)

A grid with the x axis across and the y axis up. The lines y = 2x − 1 and y = −x + 5. They cross at the dot P, the point (2, 3).Py = 2x − 1y = −x + 51234561234560xy
P is on both lines, so it fits both rules.

Worked example

Find where the line y = x + 1 crosses the line x = 3.

  1. Every point on x = 3 has x = 3.
  2. Put x = 3 into the other rule: y = 3 + 1 = 4.
  3. Check the point (3, 4) in the line x = 3. Its across number is 3.
  4. The lines cross at (3, 4).

Answer: (3, 4)

A grid with the x axis across and the y axis up. The line y = x + 1 and the upright line x = 3 cross at the dot P, the point (3, 4).Py = x + 1x = 31234561234560xy
P is on both lines. It is the point (3, 4).

See it three ways

With things

Stand on the 3 mark of a line on the ground. Walk straight forward. You stay on the 3 mark all the way.

As a picture

A grid with the x axis across and the y axis up. The upright line x = 3 with an arrow going up along it from the dot S (3, −3) to the dot F (3, 3).SFx = 3−5−4−3−2−112345−5−4−3−2−1123450xy

In symbols

x = 3 means every point has x = 3. The line is upright.

Try it

Where do two lines meet?

You need: squared paper, a ruler, a pencil.

  1. Draw the axes on squared paper, from 0 to 5.
  2. Draw the line x = 2 with a ruler. It goes up through 2 on the x axis.
  3. Draw the line y = 3 across the paper through 3 on the y axis.
  4. Mark the place where they cross. Read its two numbers.

What you will see: The lines cross at (2, 3). The upright line gives the across number and the flat line gives the up number.

If you cannot try it, this is what happens.
A grid with the x axis across and the y axis up. The upright line x = 2 and the flat line y = 3 cross at the dot P, the point (2, 3).Px = 2y = 312345123450xy

A wrong idea many people have

The line x = 3 is flat, because the x axis is flat.

Why it feels right: The x axis goes across, so a rule with x in it seems to go across too.

The test that breaks it: Plot three points with x = 3, such as (3, −2), (3, 1) and (3, 4). They are one above the other, so the line goes up.

What is true: The line x = a is upright. The line y = b is flat. Look at the letter. The rule x = 3 fixes where you stand on the x axis. So the line goes up and down.

Key words

vertical
upright, going straight up and down. The line x = 3 is vertical.
horizontal
flat, going straight across. The line y = −2 is horizontal.

Back to your guess

The answer is B: The line x = 3 is upright. It goes up and down through 3 on the x axis.

On the line x = 3 every point has x = 3, and y can be any number. The points (3, −2), (3, 1) and (3, 4) are stacked one above the other. So the line is upright.

Check yourself

Try each one before you look. A wrong answer tells you where to go next.

  1. Which line is x = 3?

    A grid with the x axis across and the y axis up. Three lines. Line A is upright, line B is flat and line C is slanted.ABC−5−4−3−2−112345−5−4−3−2−1123450xy
    Tap your answer.

    1. Not this one. Line B is flat. It is the line y = −2, not x = 3. Go to M3.9.5. Try another.

    2. Yes, that is right. Every point on x = 3 has x = 3. So the line is upright through 3 on the x axis. That is line A.

    3. Not this one. Line C slants. It is the line y = x. In x = 3 every point has x = 3. Go to M3.9.5. Try another.

    See the answer

    Line A

    Every point on x = 3 has x = 3. So the line is upright through 3 on the x axis. That is line A.

    • If you chose A: Line B is flat. It is the line y = −2, not x = 3. Go to M3.9.5.
    • If you chose C: Line C slants. It is the line y = x. In x = 3 every point has x = 3. Go to M3.9.5.

  2. The line y = −2 is drawn. What does it look like?

    Tap your answer.

    1. Not this one. An upright line comes from x = −2. The rule y = −2 fixes how far up or down. Go to M3.9.5. Try another.

    2. Not this one. It is flat, which is right. But −2 is negative, so it is below the x axis. Go to M3.9.2. Try another.

    3. Yes, that is right. Every point has y = −2. They all sit 2 below the x axis, side by side. So the line is flat.

    See the answer

    A flat line, 2 below the x axis.

    Every point has y = −2. They all sit 2 below the x axis, side by side. So the line is flat.

    • If you chose A: An upright line comes from x = −2. The rule y = −2 fixes how far up or down. Go to M3.9.5.
    • If you chose B: It is flat, which is right. But −2 is negative, so it is below the x axis. Go to M3.9.2.

  3. Which point is on the line x = 4?

    Tap your answer.

    1. Not this one. The 4 here is the up number. On x = 4 the across number must be 4. Go to M3.9.5. Try another.

    2. Not this one. The sign matters. The across number is −4, not 4. Go to M3.9.5. Try another.

    3. Yes, that is right. A point on x = 4 has an across number of 4. The up number can be anything, so (4, −1) is on it.

    4. Not this one. The sign matters. The across number is −4, not 4. Go to M3.9.5. Try another.

    See the answer

    (4, −1)

    A point on x = 4 has an across number of 4. The up number can be anything, so (4, −1) is on it.

    • If you chose A: The 4 here is the up number. On x = 4 the across number must be 4. Go to M3.9.5.
    • If you chose B: The sign matters. The across number is −4, not 4. Go to M3.9.5.
    • If you chose D: The sign matters. The across number is −4, not 4. Go to M3.9.5.

  4. The lines x = 2 and y = 5 cross. At which point?

    Tap your answer.

    1. Not this one. You swapped the numbers. The line x = 2 gives the across number. Go to M3.9.1. Try another.

    2. Yes, that is right. The crossing point is on both lines. It has x = 2 from the first line and y = 5 from the second.

    3. Not this one. The line y = 5 fixes the up number at 5, not 2. Go to M3.9.5. Try another.

    4. Not this one. The line x = 2 fixes the across number at 2, not 5. Go to M3.9.5. Try another.

    See the answer

    (2, 5)

    The crossing point is on both lines. It has x = 2 from the first line and y = 5 from the second.

    • If you chose A: You swapped the numbers. The line x = 2 gives the across number. Go to M3.9.1.
    • If you chose C: The line y = 5 fixes the up number at 5, not 2. Go to M3.9.5.
    • If you chose D: The line x = 2 fixes the across number at 2, not 5. Go to M3.9.5.

  5. Where do these two lines cross?

    A grid with the x axis across and the y axis up. The lines y = x + 1 and y = −x + 5 drawn on a grid.y = x + 1y = −x + 51234561234560xy
    Tap your answer.

    1. Yes, that is right. At (2, 3), the first rule gives 2 + 1 = 3 and the second gives −2 + 5 = 3. Both fit.

    2. Not this one. You swapped the numbers. Across comes first. Go to M3.9.5. Try another.

    3. Not this one. This point is on one line only. Put it into both rules to check. Go to M3.9.5. Try another.

    4. Not this one. This point is on one line only. Put it into both rules to check. Go to M3.9.5. Try another.

    See the answer

    (2, 3)

    At (2, 3), the first rule gives 2 + 1 = 3 and the second gives −2 + 5 = 3. Both fit.

    • If you chose B: You swapped the numbers. Across comes first. Go to M3.9.5.
    • If you chose C: This point is on one line only. Put it into both rules to check. Go to M3.9.5.
    • If you chose D: This point is on one line only. Put it into both rules to check. Go to M3.9.5.

  6. Where do the lines x = 3 and y = x + 2 cross?

    See the answer

    (3, 5)

    On x = 3 every point has x = 3. Put x = 3 into y = x + 2. Then y = 5. The lines cross at (3, 5).

  7. Where do the lines y = 4 and y = x + 1 cross?

    See the answer

    (3, 4)

    On y = 4 every point has y = 4. Put y = 4 into y = x + 1, so 4 = x + 1 and x = 3. The lines cross at (3, 4).

  8. From an earlier chapter

    Work out y when x = −2 in the rule y = 3x + 1.

    See the answer

    −5

    3 × (−2) = −6. Then −6 + 1 = −5.

Keep it

Copy this into your exercise book.

The line x = 3 is vertical, through 3 on the x axis. The line y = −2 is horizontal, through −2 on the y axis. Two lines that are not parallel cross at one point. That point fits both rules.

Draw: Draw a grid. Draw the lines x = 3 and y = −2 with a ruler. Mark where they cross and write its coordinates.

Reading with a child? A short page for the adult: what this chapter teaches and three questions to ask.

No internet tomorrow? Save the whole book as one file (761 KB). It opens on any phone.