What type of graph is it? What is the name of this course again?
(b)
Fill in the blanks: The connection between the graph and the equation above is that the graph is the collection of that the equation above.
Discovery1.2.
(a)
On the same axes as your graph for Discovery 1.1, sketch the graph of \(x+y=1\text{.}\)
(b)
Looking at your graphs, is there any pair of values \((x,y)\) that satisfy both equations simultaneously?
Discovery1.3.
(a)
On a new set of axes, sketch the graphs of \(2x+y=3\) and \(4x+2y=4\text{.}\)
(b)
Looking at these two graphs, is there any pair of values \((x,y)\) that satisfy both equations simultaneously?
Discovery1.4.
The graph of a linear equation in three variables (e.g., \(3x+y-2z=5\)) corresponds to a plane in three-dimensional space.
Suppose you had three equations in three variables. Try to imagine the geometric configuration of the corresponding three planes in each of the following situations. You might find it helpful to use three pieces of paper as props.
(a)
There is no triple of numbers \((x,y,z)\) that satisfies all three plane equations at once.
(b)
There are an infinite number of triples of numbers \((x,y,z)\) that satisfy all three plane equations at once.
(c)
There is exactly one triple of numbers \((x,y,z)\) that satisfies all three plane equations at once.
Discovery1.5.
Consider the system of equations
\begin{equation*}
\left\{\begin{array}{rcrcrcr}
x \amp + \amp 2y \amp - \amp z \amp = \amp 5, \\
\amp \amp y \amp + \amp z \amp = \amp -1.
\end{array}\right.
\end{equation*}
(a)
If \(z=2\text{,}\) what is \(y\text{?}\) … what is \(x\text{?}\)
(b)
If \(z=-10\text{,}\) what is \(y\text{?}\) … what is \(x\text{?}\)
(c)
For these example values of \(z\text{,}\) why do you think you are being asked to determine the value of \(y\) first and then to determine the value of \(x\text{?}\)
(d)
Do you think that, given any arbitrary value for \(z\text{,}\) you could solve for \(y\) and then for \(x\text{?}\)
(e)
The three values of \(x,y,z\) that you came up with in Task a together represent one solution to the system of equations. The three values of \(x,y,z\) that you came up with in Task b together represent another solution to the system of equations.
Based on your response to Task d, how many solutions does this system have in total?
(f)
If \(z=t\text{,}\) what is \(y\text{?}\) … what is \(x\text{?}\)
Discovery1.6.
Suppose \(x\) and \(y\) are “mystery” numbers, but you have a clue to their identities: you know that both \(x-2y=-4\) and \(2x+y=2\) are true.
(a)
Without determining the values of \(x\) and \(y\text{,}\) answer each of the following with a number.
(i)
\(3 (x-2y) =\) ?
(ii)
\(-2 (2x+y) =\) ?
(iii)
\((x-2y) + (2x+y) =\) ?
(iv)
\((2x+y) - 2(x-2y) =\) ?
(b)
Algebraically simplify the expression in the last part of Task a, and combine this simplified expression with your numerical answer to that part to solve for \(y\text{.}\) Then use one of the original equations from the introduction to this activity to solve for \(x\text{.}\)
Why was that combination of the left-hand sides of the two equations particularly useful for determining the values of \(x\) and \(y\text{?}\)
Discovery1.7.
We can work with a system of equations more efficiently by representing it compactly as an augmented matrix. For example,
Do you understand how this system was turned into a matrix? Now perform the following calculations, but using the matrix, obtaining a new matrix at each step.
(a)
Change the order of the equations: interchange the first and third equations.
(b)
Starting with your new system from Task a, subtract \(3\) times the first equation from the second equation, and add \(2\) times the first equation to the third equation.
(c)
Starting with your new system from Task b, multiply the second equation by \(1/3\text{.}\)
(d)
Your final result from Task c, should be a “simplified” matrix. Turn this matrix back into a system of equations and see how much easier it is to solve the system.
