Lesson 1 Row by Row Solidify Understanding

Ready

Solve each system of equations using an algebraic method.

1.

${\begin{cases} x - 2 y = 6 \\ 3 x - y = 13 \end{cases}$

2.

${\begin{cases} 2 x - 3 y = 9 \\ x + 5 y = - 28 \end{cases}$

3.

${\begin{cases} x - y = 0 \\ 4 x + y = 9 \end{cases}$

4.

${\begin{cases} 2 x + 3 y = 2 \\ - 4 x - 6 y = - 14 \end{cases}$

5.

${\begin{cases} 7 x - y = 14 \\ x + 7 y = - 48 \end{cases}$

6.

${\begin{cases} 8 x + 5 y = 9 \\ - x - 5 y = 5 \end{cases}$

7.

Do any of the systems in problems 1–6 represent parallel lines? If so, how do you know?

8.

Do any of the systems in problems 1–6 represent perpendicular lines? If so, how do you know?

Set

9.

Create a matrix to match each step in the solving of the system of equations given. Also, write a description of what happened to the equation and the matrix between steps.

System of Equations	Description	Matrix
Given System ${\begin{matrix} 3 x + 2 y = 40 \\ x - 7 y = - 2 \end{matrix}$ ⬇︎	$- 3 R_{2} \to R_{2}$ ⬇︎	$[\begin{matrix} \begin{matrix} 3 & 2 \\ 1 & - 7 \end{matrix} & \| \begin{matrix} 40 \\ - 2 \end{matrix} \end{matrix}]$ ⬇︎
Step 1 ${\begin{matrix} 3 x + 2 y = 40 \\ - 3 x + 21 y = 6 \end{matrix}$ ⬇︎	⬇︎	$[\begin{matrix} \begin{matrix} \underset{―}{} & 2 \\ - 3 & \underset{―}{} \end{matrix} & \| \begin{matrix} 40 \\ 6 \end{matrix} \end{matrix}]$ ⬇︎
Step 2 ${\begin{matrix} 3 x + 2 y = 40 \\ 0 x + 23 y = 46 \end{matrix}$ ⬇︎	⬇︎	$[\begin{matrix} \begin{matrix} \underset{―}{} & \underset{―}{} \\ 0 & \underset{―}{} \end{matrix} & \| \begin{matrix} 40 \\ \underset{―}{} \end{matrix} \end{matrix}]$ ⬇︎
Step 3 ${\begin{matrix} 3 x + 2 y = 40 \\ 0 x + y = 2 \end{matrix}$ ⬇︎	⬇︎	$[\begin{matrix} \begin{matrix} \underset{―}{} & \underset{―}{} \\ \underset{―}{} & \underset{―}{} \end{matrix} & \| \begin{matrix} \underset{―}{} \\ \underset{―}{} \end{matrix} \end{matrix}]$ ⬇︎
Step 4 ${\begin{matrix} 3 x + 0 y = 36 \\ 0 x + y = 2 \end{matrix}$ ⬇︎		$[\begin{matrix} \begin{matrix} \underset{―}{} & \underset{―}{} \\ \underset{―}{} & \underset{―}{} \end{matrix} & \| \begin{matrix} \underset{―}{} \\ \underset{―}{} \end{matrix} \end{matrix}]$ ⬇︎
Step 5 ${\begin{matrix} x + 0 y = 12 \\ 0 x + y = 2 \end{matrix}$		$[\begin{matrix} \begin{matrix} \underset{―}{} & \underset{―}{} \\ \underset{―}{} & \underset{―}{} \end{matrix} & \| \begin{matrix} \underset{―}{} \\ \underset{―}{} \end{matrix} \end{matrix}]$

Go

10.

Flip a coin $5$ times. Record the number of times the coin lands with heads up. Repeat this process $20$ times, either by hand or by simulation using technology, each time recording your results in the table below. (Every time you perform the simulation, count the number of heads you have and record the result in the Tally column below. For example, if you flip the coin $5$ times and get $3$ heads, put a tally mark by the $3$ heads or $60 %$ row.)

https://openup.org/itgIdg

If using this simulation click “draw samples” button after you have entered the number of tosses.

# Heads	% Heads	Tally
$0$	$0 %$
$1$	$20 %$
$2$	$40 %$
$3$	$60 %$
$4$	$80 %$
$5$	$100 %$

11.

Create a histogram of your results. Describe the shape of the histogram (Shape, Center, Spread)

12.

Flip a coin $20$ times. Record the number of times heads lands up out of the $20$ flips. Repeat this process $20$ times either by hand or by simulation using technology.

https://openup.org/itgIdg

Record your results in the table provided.

# Heads	% Heads	# Heads	% Heads
$0$	$0 %$	$11$	$55 %$
$1$	$5 %$	$12$	$60 %$
$2$	$10 %$	$13$	$65 %$
$3$	$15 %$	$14$	$70 %$
$4$	$20 %$	$15$	$75 %$
$5$	$25 %$	$16$	$80 %$
$6$	$30 %$	$17$	$85 %$
$7$	$35 %$	$18$	$90 %$
$8$	$40 %$	$19$	$95 %$
$9$	$45 %$	$20$	$100 %$
$10$	$50 %$

13.

Create a histogram of your results. Describe the shape of the histogram (Shape, Center, Spread)

14.

Compare the shape center and spread of each distribution. What do you notice?

15.

If you repeated this process with $500$ flips instead of $5$ or $20$ , predict what would happen to the shape, spread, and center of the new histogram.