[size=150]Write a few equations that are equivalent to equation A by multiplying both sides of it by the same number, for example, 2, -5, or [math]\frac{1}{2[/math]. Let’s call the resulting equations A1, A2, and A3. Record your equations here:[br][/size][br]Equation A1:

Equation A2:[br]

Equation A3:[br]

[center][math]\begin {cases}\begin {align} 4x + \hspace{2.2mm} y &= \hspace {2mm}1 &\quad&\text{Equation A}\\ x + 2y &= \hspace {2mm} 9&\quad&\text{Equation B} \end{align} \end{cases}[/math][/center][br]To start solving the system, Elena wrote: [br][center][math]\begin {align} 4x + \hspace{2.2mm} y &= \hspace {2mm}1\\ 4x + 8y &= 36 \end{align}[/math][/center][br]And then she wrote: [br][center][math]\begin {align} 4x + \hspace{2.2mm} y &= \hspace {3mm}1\\ 4x + 8y &= \hspace{1mm}36 \hspace{1.5mm}- \\ \overline {\hspace{8mm}\text-7y} &\overline{\hspace{1mm}=\text-35 \hspace{5mm}}\end{align}[/math][/center]What were Elena's first two moves?

What might be possible reasons for those moves?

Complete the solving process algebraically. Show that the solution is indeed [math]x=-1,y=5[/math].[br]

[math]\begin {cases}\frac45 x + 6y = 15\\ \text-x + 18y = 11 \end{cases}[/math][br][br]Arrange the slips in the order that would lead to a solution. Be prepared to:[br][list][*]Describe what move takes one system to the next system.[/*][*]Explain why each system is equivalent to the one before it[/*][/list]

[math]\begin {cases}Ax - By = 24\\ Bx + Ay = 31 \end{cases}[/math][br][br]Find the missing values [math]A[/math] and [math]B[/math].

[center][math]\begin {cases} \begin {align}12a + 5b &= \text-15\\8a + \hspace{2mm}b &= \hspace{1.5mm}11 \end{align} \end {cases}[/math][/center][br]To solve this system, Diego wrote these equivalent systems for his first two steps.[br][table][tr][td]Step 1:[/td][td]Step 2:[/td][/tr][tr][td][math]\begin {cases} \begin {align}12a + \hspace{1.5mm}5b &= \text-15\\\text-40a + \text-5b &= \text-55 \end{align} \end {cases}[/math][/td][td][math]\begin {cases} \begin {align}12a + 5b &= \text-15\\\text-28a \hspace{8.5mm}&= \text-70 \end{align} \end {cases}[/math][/td][/tr][/table][br]Describe the move that Diego made to get each equivalent system. Be prepared to explain how you know the systems in Step 1 and Step 2 have the same solution as the original system.

Write another set of equivalent systems (different than Diego's first two steps) that will allow one variable to be eliminated and enable you to solve the original system. Be prepared to describe the moves you make to create each new system and to explain why each one has the same solution as the original system.[br]

Use your equivalent systems to solve the original system. Then, check your solution by substituting the pair of values into the original system. [br]

[math]\begin{cases} 2x-4y=10 \\ x+5y=40 \\ \end{cases}[/math]

[math]\begin{cases} 3x-5y=4 \\ \text-2x + 6y=18 \\ \end{cases}[/math]

[size=150][math]\begin{cases} 4p+2q=62 \\ 8p-q=59 \\ \end{cases}[/math][br]He can think of two ways[/size][size=150] to eliminate a variable and solve the system:[/size][br][list][*]Multiply [math]4p+2q=62[/math] by 2, then subtract [math]8p-q=59[/math] from the result.[/*][*]Multiply [math]8p-q=59[/math] by 2, then add the result to [math]4p+2q=62[/math].[/*][/list][br]Do both strategies work for solving the system? Explain or show your reasoning. 

[math]\displaystyle \begin{cases} y=3x \\ y=9x-30 \end{cases}[/math][br][list][*][size=150]Andre's first step is to write: [/size][math]3x=9x-30[/math][/*][size=150][*]Elena’s first step is to create a new system: [math]\displaystyle \begin{cases} 3y=9x \\ y=9x-30 \end{cases}[/math][/*][/size][/list][br]Do you agree with either first step? Explain your reasoning. 

[math]\begin{cases}\begin{align} 6d+4.5e&=16.5\\5d+0.5 e&=\hspace{2mm}4 \end{align}\end{cases}[/math]

[math]\begin{cases}\begin{align} p+8q&=\text-8\\ \frac12p+5q&=\text-5 \end{align}\end{cases}[/math][br][br]Write a system of equations that is equivalent to this system. Describe what you did to the original system to get the new system.[br]

Explain how you know the new system has the same solution as the original system.[br]

[list][*]Write an equation that relates the number of postcard stamps [math]p[/math], the number of first-class stamps [math][/math]f, and the cost of mailing the package.[/*][*]Solve the equation for [math]f[/math].[/*][/list]

[list][*]Solve the equation for [math]p[/math].[/*][/list]

If Noah puts 7 first-class stamps on the package, how many postcard stamps will he need?

[math]\begin{cases} 2x+7y=8 \\ y+2x=14 \ \end{cases}[/math][br][br]Find at least one way to solve the system by substitution and show your reasoning. How many ways can you find? (Regardless of the substitution that you do, the solution should be the same.)

[math]\begin{cases} \text-7x + 3y= \text-65 \\ \text -7x+ 10y= \text-135 \\ \end{cases}[/math][br][br]Write an equation that results from subtracting the two equations.

[size=150] Priya buys 2.2 pounds of bananas and 3.6 pounds of grapes for $9.35. Andre buys 1.6 pounds of bananas and 1.2 pounds of grapes for $3.68.[br]This situation is represented by the system of equations:[/size][br][math]\begin{cases} 2.2b + 3.6g = 9.35 \\ 1.6b + 1.2g = 3.68 \\ \end{cases}[/math][br][br]Explain why it makes sense in this situation that the solution of this system is also a solution to [math]3.8b+4.8g=13.03[/math].