Linear - determine equations TYS 1

Where no diagram has been provided by the cruel person writing these questions,
draw your own set of axes so you can get a feeling for the problem and therefore for the equation.

Interpreting basic equations.

1. What are the x and y intercepts for the line whose equation is y = 3x - 2?

2. What are the x and y intercepts for the line whose equation is y = -2x + 3?

3. What are the x and y intercepts for the line whose equation is 2x + 3y - 2 = 0?

4. What are the x and y intercepts for the line whose equation is 4x - 5y - 1 = 0?

Given 1 point and the gradient
(answer in gradient intercept form).

5. Determine the equation of the line through the point (1,2) with a gradient of 5.
Answer.y = 5x - 3.

6. Determine the equation of the line shown in the graph below:

Answer.y = x.

7. Determine the equation of the line through the point (-1, -2) with a gradient of -4.
Answer.y = -4x - 6.

8. Determine the equation of the line shown in the graph below:

Answer.y = -2x + 2.

9. Determine the equation of the line through the point (1, -2) with a gradient of 0.
Answer.y = -2.

Given two points
(answer in general form).

10. Determine the equation of the line shown in the graph below:

11. Determine the equation of the line through the points (10, 3) and (15, 8).
Answer.y = x - 7.

12. Determine the equation of the line through the points (-12, 3) and (4, -5)

13. Determine the equation of the line through the points (3.5, 2.7) and (5.5, 10.7).
Answer.y = 4x - 11.3.

14. Determine the equation of the line through the points (3, -2) and (3, 2).
Answer.x = 3.

Given data in a table.

15. The following table shows data based on a linear relationship between A and B.

X	-1	0	1	15	??
Y	-3	-1	1	??	95

(i) Determine the linear equation describing these data points.

(ii) What are the values for the two missing entries?

Answer.(i) B = 2A -1.
(ii) Missing numbers are Y = 29 and X = 48.

16. The following table shows data based on a linear relationship between X and Y.

X	4	5	6	7	??
Y	7	3	-1	-5	-33

(i) Determine the linear equation describing these data points.

(ii) What is the value missing in the top row?

Answer.(i) Y = -4X + 23.
(ii) Missing number is X = 14.

Given data in a diagram.

17. What is the equation of the blue line shown in the following diagram?

Answer.y = 6 - 2x.

18. What is the equation of the blue line shown in the following diagram?

Answer.y = x/3 - 3 or
x - 3y - 9 = 0.

Given a geometrical context.

19. What is the equation of the horizontal line passing through the point (-22, 15)?
Answer.y = 15.

20. What is the equation of the vertical line passing through the point (26, -26)?
Answer.x = 26.

21. A is the point (2, 3) and the point C lies on the y axis.

The equation of the line AC is 3x - y - 9 = 0.

Given ABCD is a rectangle, determine the coordinates of the points B and D.

Answer.B is (0, 3) and D is (2, -9).

22. The vertex of the isosceles triangle ABC is the point A with coordinates (3, 4). B is the point (0, 7) and the base of the triangle lies along the y axis.

Draw a sketch showing this information and hence find the coordinates of point C.

Answer.C is (0, 1).

Applications to the physical world.

23. Ish starts to save her money in a special holiday account. She begins by depositing $500 and then deposits $40 each week from her part time job.

(i) Draw a graph of her investment amount against time.

(ii) Develop an equation to link Ish's total investment against time.

(iii) How much will Ish have in her account at the end of 26 weeks (assuming she does not withdraw any money during that time).

Answer.Ish saves $1,540 after 26 weeks.

24. A car is stationary at a given time. It then moves and increases its speed at a constant rate of 0.4 metres/second.

(i) Draw a graph showing the change in the car's speed against time.

(ii) Develop an equation to the car's speed against time.

(iii) Use your equation to estimate how long it will take to reach a speed of about 16.7 m/sec (equivalent to a speed of about 60 kph)?

Answer.It takes 41.75 secs.

25. The average height, C, in centimetres, of a girl between the ages of 6 years and 11 years can be represented by a line with the equation C = 6A + 79 where A is the age of a girl in years.

(i) How high would we expect an 8 year old girl to be ?

(ii) What is the value of the gradient of this line and what does the value for the gradient indicate about the changing height of girls aged 6 to 11?

(iii) Give ONE reason why this equation is not suitable for predicting heights of girls older than 12.