**Determining whether a function is linear :**

The equations which are given in the form ax + by + c = 0 or y = mx + b are considered to be linear equations. The linear equations represent linear functions. Equations that cannot be written in the above forms form are not linear equations, and therefore are not linear functions.

**Example 1 :**

A square tile has a side length of x inches. The equation aaa y = x² gives the area of the tile in square inches. Determine whether the relationship between x and y is linear and, if so, if it is proportional.

**Solution : **

**Step 1 :**

Choose several values for the input x. Plug these values for x in the equation to find the output y.

**x**

1

2

3

4

**x²**

1²

2²

3²

4²

**y**

1

4

9

16

**(x, y)**

(1, 1)

(2, 4)

(3, 9)

(4, 16)

**Step 2 :**

Graph the ordered pairs. Then draw a line through the points to represent the solutions of the function.

**Step 3 : **

Describe the relationship between x and y.

The graph is not a line. So the relationship between the side length of the square and its area is non linear.

Only a linear relationship can be proportional. Since the relationship is non linear, it is not proportional.

**Example 2 :**

The temperature of a fish tank was 82 °F and decreased steadily 2 °F every hour. The equation y = -2x + 82 gives the temperature y after x hours. State whether the relationship between the time and the temperature is linear, if so, if it is proportional.

**Solution : **

**Step 1 :**

Choose several values for the input x. Plug these values for x in the equation to find the output y.

**x**

0

1

2

3

4

5

**-2x + 82**

-2(0) + 82

-2(1) + 82

-2(2) + 82

-2(3) + 82

-2(4) + 82

-2(5) + 82

**y**

82

80

78

76

74

72

**(x, y)**

(0, 82)

(1, 80)

(2, 78)

(3, 76)

(4, 74)

(5, 72)

**Step 2 :**

Graph the ordered pairs. Then draw a line through the points to represent the solutions of the function.

**Step 3 :**

Describe the relationship between x and y.

The graph is a line. So the relationship between the time and the temperature is linear.

But the line is not passing through the origin. So, it is not proportional.

**Note : **

The graphs of all linear relationships will be a line. If the line passes through the origin, the linear relationship is proportional.

If the line does not pass through the origin, the linear relationship is non proportional.

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