COLLEGE ALGEBRA: AN INVESTIGATIVE APPROACH

by

S. Kastberg & T. Barron


LINEAR SYSTEMS


Please type your answers to the questions in the lesson in the text boxes below. Please type all your answers in complete sentences. After you have entered all your answers please be sure to click on the submit button at the end of the lesson.


1. Graph the lines y = 2x + 1 and y = 2x - 1 HERE or using a computer graphing tool such as Graphing Calculator .

Find the slope and the y-intercept of each line.

y = 2x + 1 m= b=

y = 2x - 1 m= b=

Number of intersections

Points of intersection

2. Graph the lines y = 2x + 1 and y = -3x - 1 HERE or using a computer graphing tool.

Find the slope and the y-intercept of each line.

y = 2x + 1 m= b=

y = -3x - 1 m= b=

Number of intersections

Points of intersection

3. Graph the lines y = 2x + 1 and 2y = 4x + 2 HERE or using a computer graphing tool.

Find the slope and the y-intercept of each line.

y = 2x + 1 m= b=

2y = 4x + 2 m= b=

Number of intersections

Points of intersection

4. Look at the previous three examples and come up with a general rule about the number of intersections possible for two lines, given the slope and the y-intercept.

a. If the slope and the y-intercept of two lines are the same the lines have point(s) of intersection.

b. If the slopes of two lines are the same and the y-intercepts are different, the lines have point(s) of intersection.

c. If the slopes of the two lines are different, regardless of the y-intercepts, the lines have point(s) of intersection.

Solving Systems of Equations Algebraicly

5. Graph each of the following equations HERE or using a computer graphing tool.

a. 2y + 3x = 5

x-intercept = y-intercept =

b. -4y + x = 2

x-intercept = y-intercept =

6. Estimate the point of intersection of these two lines.

The equations 2y + 3x = 5 and -4y + x = 2 form a linear system. What happens if you change the equations a bit?

7. If I multiply 2y + 3x = 5 by 2 what is the resulting equation .

8. Graph 2y + 3x = 5 and your answer to question seven on the same axes HERE or using a computer graphing tool.

What do you observe?

 

9. Do you see any similarities between the equation you found in question seven and the equation -4y + x = 2? If so what are they?

10. Please add the equation you found in question seven and the equation -4y + x =2.

What is the resulting equation?

What variable has been eliminated from the equations?

11. Solve the equation you found in question 10.

12. Compare the answer you found in question 11 to the x-coordinate of the estimated intersection you found in question 6. Are they close? If they are why do you think they are, if not why not?

13. Now that we know the x coordinate of the point that is on both lines (2y + 3x = 5 and -4y + x = 2) explain how we could find the y-coordinate of that point.

14. Find the y-coordinate.

15. Is it close to the estimate you made in question 6?

16. The point of intersection of 2y + 3x = 5 and -4y + x = 2 is .

17. So why did we multiply 2y + 3x = 5 by 2 in the system 2y + 3x = 5 ; -4y + x = 2?

18. Given the following system what might you multiply the first equation by to eliminate one of the variables when the equations are added?

a. 2x - y = -3

b. 3x + 5y = 1

Multiply equation a by .

19. Given the following system what might you multiply the first equation by to eliminate one of the variables when the equations are added?

a. 2x - 4y = -3

b. 2x + 5y = 1

Multiply equation a by .

20. Solve the following system. Be sure to graph the system to see if your answer makes sense. Click HERE to graph or use a graphing tool on your computer.

a. 2x - 4y = -3

b. x - 4y = 1

The solution to the above system is .


To submit your responses click on the submit button

To reset the form click on the reset button


Return