A thorough understanding of a system of linear equations

This app contains two lines in the general forms as follows:[br][br][math]a_1x+b_1y=c_1[/math] and [math]a_2x+b_2y=c_2[/math].[br][br]Experiment with the sliders for [math]a_1[/math], [math]b_1[/math], [math]c_1[/math], [math]a_2[/math], [math]b_2[/math], and [math]c_2[/math], to answer the following questions.

1. What is the slope of the first line (red line) when a1 = 2, b1 = 3?

Cannot determine. We need c1 2/3 None of the above -2/3

2. Experiment with the sliders and answer when will both the lines coincide?

When a1 = a2, b1 is not equal to b2 and c1 is not equal to c2. When a1 = a2, b1 = b2 and c1 is not equal to c2. When a1 / a2 = b1 / b2 = c1/c2 When a1 is not equal to a2, b1 = b2 and c1 is not equal to c2. When a1 / a2 = b1 / b2 but is not equal to c1/c2 When a1 = a2, b1 = b2 and c1 = c2

3. What happens to the graph of both the lines when a1 = a2, b1 = b2 but c1 and c2 are not equal?

Cannot determine Both the graphs will be parallel. Both the graphs will have a unique intersection point Both the graphs will coincide.

4. Set a1 = 4, b1 = 2 and c1 = 1. What will be the value of a2 + b2 + c2 when the black line is 4 units below the red line?

1 2 0 -2 -1

5. Set a1 = k(a2), b1 = k(b2) and c1 = k(c2) (here k can be any positive integer). For example, a1= 2, a2 = 4, b1 = -1, b2 = -2, c1 = 1 and c2 = 2. [br][br]Experiment with the sliders as required and select the correct answer choices from below.

After a1 = k*(a2) and b1 = k*(b2) are set (using your own choices of a1, b1, a2, b2 and k), changing c1 and c2 will make the lines intersect each other. After a1 = k*(a2) and b1 = k*(b2) are set (using your own choices of a1, b1, a2, b2 and k) changing c1 and c2 will make the lines parallel. After a1 = k*(a2) and b1 = k*(b2) are set (using your own choices of a1, b1, a2, b2 and k) changing a1 or a2 will make the lines intersect. After a1 = k*(a2) and b1 = k*(b2) are set (using your own choices of a1, b1, a2, b2 and k) changing c1 or c2 will make the lines move up or down with respect to each other's position. After a1 = k*(a2) and b1 = k*(b2) are set (using your own choices of a1, b1, a2, b2 and k) changing b1 or b2 will make the lines intersect.

A thorough understanding of a system of linear equations

Information: A thorough understanding of a system of linear equations