# EXAMINING LINEAR RELATIONSHIPS

Examining linear relationships :

Recall that a proportional relationship is a relationship between two quantities in which the ratio of one quantity to the other quantity is constant.

The graph of a proportional relationship is a line through the origin.

When ratios between quantities are not constant, a relationship may be linear but not proportional and the graph does not pass through the origin.

## Examining linear relationships - Example

The entrance fee for Mountain World theme park is \$20. Visitors purchase additional \$2 tickets for rides, games, and food. The equation y = 2x + 20 gives the total cost, y, to visit the park, including purchasing x tickets.

Explain why the relationship between number of tickets and total cost is not proportional.

Solution :

Step 1 :

Choose several values for x that make sense in context.

x  =  1, 2, 3, 4

Step 2 :

Use the equation y = 2x + 20 to find y for each value of x.

x  =  1 :

y  =  2(1) + 20

y  =  2 + 20

y  =  22

x  =  2 :

y  =  2(2) + 20

y  =  4 + 20

y  =  24

x  =  3 :

y  =  2(3) + 20

y  =  6 + 20

y  =  26

x  =  4 :

y  =  2(4) + 20

y  =  8 + 20

y  =  28

Step 3 :

Let us list out the values of y for the corresponding values of x using a table.

Step 4 :

Plot the ordered pairs from the table. Describe the shape of the graph.

The points lie on a line.

Step 5 :

Find the rate of change between each point and the next. Is the rate constant ?

Yes, the rate is constant. It is \$2 per ticket.

Step 6 :

Calculate y/x for the values in the table.

Explain why the relationship between number of tickets and total cost is not proportional.

Undefined, 12, 7, about 5.3, 4.5 ;

The ratio of the total cost to the number of tickets sold is not constant, and the graph doesn’t pass through the origin.

## Reflect

1.  Would it make sense to add more points to the graph from x = 0 to x = 10 ?

Yes

We could add the points (1, 22), (3, 26), (5, 30), (7, 34), (9, 38), and (10, 40).

2.  Would it make sense to connect the points with a line ? Explain.

No

Because, the number of tickets can only be a whole number.

After having gone through the stuff given above, we hope that the students would have understood, how to examine linear relationships.

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