Question

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Determine the slope and the y-intercept of the linear function given in the table: x 1 3 | 6 8 | 11 | y -0.5 0.5 2 3 4.5 slope = slope = O Preview Preview y-intercept =

Answer 1

Q1.

Let the linear function be y = a × x + b. Here, a is the slope and b is the y-intercept. It is given that,

at x = 1, y = -0.5 = a × 1 + b = a+b .....1

at x = 3, y = 0.5 = a×3 + b = 3a+b .......2

We need just 2 points as the number of unknowns are two. The rest of the points can be used to verify our result.

Adding equation 1 and 2, we get

0 = 4a + 2b

or b = -2a

Putting b = -2a in equation 2, we get

0.5 = 3a - 2a = a

Hence, a = 0.5 and b = -2a = -1.

Thus the linear function is y = 0.5x - 1. Hence, slope = 0.5 and intercept = -1.

Q2.

The line intercepts the y-axis at (0,3) and the x-axis at (4.5,0). The slope can be calculated by two methods. One is following the approach in the previous question. Another is using the definition of slope of a line.

Slope = m = (y1-y2)/(x1-x2) ; where (x1,y1) and (x2,y2) are any two points on the line.

Thus m = (3-0)/(0-4.5) = 3/(-4.5) = -2/3

Q3.

Using the same approach as Q2, the line intercepts the y-axis at (0,4) and the x-axis at (1,0). Now we have the required two points for slope calculation.

m = (4-0)/(0-1) = 4/(-1) = -4

Q4.

It is similar to Q1. Let the relationship be y = ax + b.

at x = 1, y = 8.5 = a × 1 + b = a+b .....1

at x = 3, y = 11.5 = a×3 + b = 3a+b .......2

Subtracting equation 1 from 2, we get

11.5 - 8.5 = 3a+b-a-b  

or 3 = 2a

or a = 3/2

Putting a = 3/2 in equation 1, we get

8.5 = 1.5 + b

or b = 7.  

Thus the linear function is y = (3/2)x + 7. Hence, slope = 3/2 and y-intercept = 7.