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Discovery guide 1.1 Discovery guide
Discovery 1.1 .
Sketch the graph of
\(2x+y=3\text{.}\)
(a) 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.
Discovery 1.2 .
(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?
Discovery 1.3 .
(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?
Discovery 1.4 .
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.
Discovery 1.5 .
Consider the system of equations
\begin{equation*}
\begin{sysofeqns}{rcrcrcr}
x \amp + \amp 2y \amp - \amp z \amp = \amp 5, \\
\amp \amp y \amp + \amp z \amp = \amp -1.
\end{sysofeqns}
\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
\(x\text{?}\)
To solve a systems of equations, we will manipulate and combine the equations in the system in both familiar and (possibly) unfamiliar ways.
Discovery 1.6 .
Suppose
\(x\) and
\(y\) are “mystery” numbers, but you have a clue to their identities: you know that both
\(x - 2 y = -4 \) and
\(2 x + y = 2 \) are true.
(a)
Without determining the values of
\(x\) and
\(y\text{,}\) fill in each blank with a
number .
(i)
\(\displaystyle
\quad
\begin{array}{rcl}
x - 2 y \phantom{)} \amp = \amp -4 \\ \hline
3 (x - 2 y) \amp = \amp \fillinmath{XX}
\end{array}\)
(ii)
\(\displaystyle
\quad
\begin{array}{rcl}
2 x + y \phantom{)} \amp = \amp \phantom{-} 2 \\ \hline
-2 (2 x + y) \amp = \amp \fillinmath{XX}
\end{array}\)
(iii)
\(\displaystyle
\quad
\begin{array}{rcl}
x - 2 y \phantom{)} \amp = \amp -4 \\
2 x + y \phantom{)} \amp = \amp \phantom{-} 2 \\ \hline
(x - 2 y) + ( 2 x + y) \amp = \amp \fillinmath{XX}
\end{array}\)
(iv)
\(\displaystyle
\quad
\begin{array}{rcl}
2 x + y \phantom{)} \amp = \amp \phantom{-} 2 \\
x - 2 y \phantom{)} \amp = \amp -4 \\ \hline
(2 x + y) - 2 (x - 2 y) \amp = \amp \fillinmath{XX}
\end{array}\)
(b) Algebraically simplify the left-hand side of the last equation in
Task a.iv , and use your simplified new equation 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 two equations particularly useful for determining the values of
\(x\) and
\(y\text{?}\)
Now we’ll use the lessons learned in
Discovery 1.6 to simplify systems of equations.
Discovery 1.7 .
We can work with a system of equations more efficiently by representing it compactly as an augmented matrix . For example,
\begin{equation*}
\begin{sysofeqns}{rcrcrcr}
-2x \amp + \amp 2y \amp - \amp 5z \amp = \amp -1 \\
3x \amp \amp \amp + \amp 3z \amp = \amp 9 \\
x \amp - \amp y \amp + \amp 3z \amp = \amp 2
\end{sysofeqns}
\quad\longrightarrow\quad
\begin{abmatrix}{rrr|r}
-2 \amp 2 \amp -5 \amp -1 \\
3 \amp 0 \amp 3 \amp 9 \\
1 \amp -1 \amp 3 \amp 2
\end{abmatrix}
\end{equation*}
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.
