Connections Among Proportional Relationships, Lines & Linear Equations

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

Proportional relationships in mathematics happen when two values always change by the same multiple. If one doubles, the other doubles. This lesson will discuss the connection among proportional relationships, lines, and linear equations.

What are Proportional Relationships?

''Mom, I have a cake recipe, but I need a cake that's twice as big. What should I do?''

''It's proportional, Laticia. If you're doubling the size of the cake, then you'll just double all of the amounts of all the ingredients.''

Proportional relationships occur when two variables always change by the same multiple. If one doubles, the other doubles. If one is cut in half, the other will also be halved.

Linear Equation of a Proportional Relationship

You can always reduce a proportional relationship to the same equation:

y = k x

This means that with proportional relationships:

  • x may be multiplied by k to find y
  • y may be divided by k to find x
  • k may be found by dividing y by x

For example, if we have the proportional relationship y = 2x, then:

  • k = 2
  • every x gets multiplied by 2 to produce y
  • if we feed in a 4 for x, we'll get an 8 (4 times 2) for y
  • any value for y could be divided by 2 to get to x
  • if y is 6, then dividing it by 2 would mean that x is 3

If we had a proportional relationship where we didn't know what k was, but we knew that y was 15 when x was 5, then we could divide 15 by 5 to get 3 for k.

Identifying Graphs of Proportional Relationships

Because the two variables always change by the same multiple, the graph of a proportional relationship is always:

  • a straight line (no curves, bends, corners, or open spots)
  • a line that passes through the origin (the 0,0 intersection)
  • not vertical or horizontal (doesn't look like a floor or a wall)

The above graph is a proportional relationship because it is a straight line and passes through (0,0).
image Proportional Graph

This graph is not proportional because it does not pass through (0,0).
image Non-proportional Graph

This graph is not proportional because it is not a straight line.
image Non-proportional Graph 2

Drawing a Graph of a Proportional Relationship

To graph a proportional relationship, you would:

  1. Put one point at (0,0), where the x-axis and the y-axis meet.
  2. Find the constant k.
  3. Pick a value on the x-axis, such as 1, 2, 5, etc.
  4. Multiply that x value by the constant k to get a y value.
  5. Plot your new point on the graph, going across to the x value, then up (or down) to the y value.
  6. Draw a straight line through the (0,0) point and your new point.

For example, if you have the equation y = 3x, you could:

  1. Grab your graph paper.
  2. Draw your x and y axes.
  3. Place a dot at the point where the two axes meet (the origin at 0,0).
  4. Pick a value to use for x, such as 2.
  5. Multiply that number by 3, which is your k for this problem. If you picked 2, then the y value will be 6.
  6. Put a dot at the (2,6) intersection.
  7. Draw a straight line through (0, 0) and the (2,6) point.

That's all there is to it!

Proportional graph passing through (2,6)
image Drawing a Proportional Graph

Example of a Proportional Relationship

Space Shuttle Atlantis
space shuttle graphic

A common proportional relationship is the one between how long you travel at a certain speed and how far you go:

d = r t

  • d is the distance traveled (how far you went)
  • r is the rate of speed (how fast you're going)
  • t is the time traveled (how long you've been going that fast)

For example, imagine you're in a space shuttle going at 17,500 miles per hour, circling the Earth, and you've been going for two hours. How far have you traveled?

Well, you know that distance traveled equals how long you've traveled times the speed you're going, so after grabbing your calculator, you punch in 17500 and multiply it by 2. You've gone 35,000 miles in those two hours!

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