Answers to Tutorial on Systems Of Equations

Answers to the questions in the tutorial of Systems of Linear Equations are presented.

Question:

Use the scrollbar to set the constants a1, a2, b1, b2, c1 and c2 so that the two lines have a point of intersection (x , y ). Check that the ordered pair (x , y), is the solution to the system (find a solution by any method you wish : substitution, elimination, cramer's rule,...) and compare.

Answer:

As an example set a1 = 1, b1 = 2, c1 = 2, a2 = 1, b2 = 1 and c2 = 0. The sytems of equations is given by:

x + 2y = 2
x + y = 0
For these values of the coefficients, the point of intersection is at (-2 , 2). If x and y in the above systems of equations are substituted by -2 and 2 respectively, both equations are satisfied.
Solve by substitution: the second equation gives y = - x. Substitute y in the first equation by -x, we obtain
x + 2(-x) = 2
Solve the above equation for x, you obtain x = -2. y = -x gives y = 2. So the analytical solution to the system is (-2 , 2) which is exactly the solution found graphically.


Question:

Set a1, a2, b1 and b2 to values such that a1*b2 - a2*b1 = 0. How are the two lines positioned with respect to each other ? How many solutions the system has ? If (the determinant) a1*b2 - a2*b1 = 0, what is the relationship between the slopes of the two lines (are they equal for example?) ?

Answer:

An an example set a1 = 1, b1 = 1, c1 = 0, a2 = 3, b2 = 3 and c2 = 6. You can check that a1*b2 - a2*b1 = 3 - 3 = 0. The two lines are parallel. The system has no solutions, in general.
If a1*b2 - a2*b1 = 0, then a1/b1 = a2/b2. a1/b1 is the solpe of the line a1*x + b1*y = c1 and a2/b2 is the solpe of the line a2*x + b2*y = c2. So a1*b2 - a2*b1 = 0 means that the two lines have the same slope and are parallel. the two lines have no point of intersection in this case.


Question:

Set all 6 constants to values such that a1/a2 = b1/b2 = c1/c2. How many solutions the system has ?Explain.

Answer:

An an example set a1 = 2, b1 = 2, c1 = 2, a2 = 1, b2 = 1 and c2 = 1. You can check that a1/a2 = b1/b2 = c1/c2 = 2. The two equations are given by:

x + y = 1

2x + 2y = 2
Although the equations are different, they are equivalent. Take the first equation and multiply all terms by 2, you obatin the second equation. That is why they give the same graph. The system has an infinite number of solutions.

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