Solving Systems by Elimination (Part 3)
10 min
Teacher Prep
Required Preparation
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if students each have their own device. (Desmos is available under Math Tools.)
Narrative
In this Warm-up, experiment with the graphical effects of multiplying by a factor both sides of an equation in two variables.
Students are prompted to multiply one equation in a system by several factors to generate several equivalent equations. They then graph these equations on the same coordinate plane that shows the graphs of the original system. Students notice that no new graphs appear on the coordinate plane and reason about why this might be the case.
The work here reminds students that equations that are equivalent have all the same solutions, so their graphs are also identical. Later, students will rely on this insight to explain why we can multiply one equation in a system by a factor—which produces an equivalent equation—and solve a new system containing that equation instead.
Launch
Display the equation x+5=11 for all to see. Ask students:
- "What is the solution to this equation?" (x=6)
- "If we multiply both sides of the equation by a factor, say, 4, what equation would we have?" (4x+20=44) "What is the solution to this equation?" (x=6)
- "What if we multiply both sides by 100?" (100x+500=1,100) "By 0.5?" (0.5x+2.5=5.5)
- "Is the solution to each of these equations still x=6?" (Yes)
Remind students that these equations are one-variable equations, and that multiplying both sides of a one-variable equation by the same factor produces an equivalent equation with the same solution.
Ask students: "What if we multiply both sides of a two-variable equation by the same factor? Would the resulting equation have the same solutions as the original equation?" Tell students that they will now investigate this question by graphing.
Arrange students in groups of 2–4, and provide access to graphing technology to each group. To save time, consider asking group members to divide up the tasks. (For example, one person could be in charge of graphing while the others write equivalent equations, and all analyze the graphs together.)
Student Task
Consider two equations in a system:
\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}
- Use graphing technology to graph the equations. Then, identify the coordinates of the solution.
-
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 21. Let’s call the resulting equations A1, A2, and A3. Record your equations here:
- Equation A1:
- Equation A2:
- Equation A3:
- Graph the equations that you generated. Make a couple of observations about the graphs.
Sample Response
- Sample response:
- Equation A1: 8x+2y=2
- Equation A2: -20x−5y=-5, or 20x+5y=5
- Equation A3: 2x+21y=21
- Sample response: No new graphs appeared. The graphs of the equivalent equations are identical to that of Equation A. The graphs of Equations A1, A2, and A3 all intersect the graph of Equation B at (1,-5).
Activity Synthesis (Teacher Notes)
Invite students to share their observations about the graphs they created. Ask students why the graphs of Equations A1, A2, and A3 all coincide with the graphs of the original Equation A. Discuss with students:
- "How can we explain the identical graphs?" (Equations A1, A2, and A3 are equivalent to Equation A, so they all have the same solutions as Equation A, and their graphs are the same line as the graph of Equation A.)
- "What move was made to generate Equations A1, A2, and A3? Why did it create equations that are equivalent to Equation A?" (The two sides of Equation A were multiplied by the same factor, which keeps the two sides equal.)
To further illustrate that Equations A1, A2, and A3 are equivalent to Equation A, and if time permits, consider:
- Using tables to visualize the identical (x,y) pairs. For example:
4x+y=1
x y 0 1 1 -3 2 -7 3 -11 20x+5y=5
x y 0 1 1 -3 2 -7 3 -11 2x+21y=21
x y 0 1 1 -3 2 -7 3 -11 - Remind students that, earlier in the unit, they saw that isolating one variable is a way to see if two equations are equivalent. If we isolate y in Equations A, A1, A2, and A3, the rearranged equations will be identical: y=-4x+1.
Standards
- A-REI.1·Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
- A-REI.1·Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
- A-REI.1·Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
- HSA-REI.A.1·Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
- A-REI.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
- A-REI.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
- A-REI.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
- HSA-REI.C.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
15 min
10 min
Knowledge Components
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Solving Systems by Elimination (Part 3)
10 min
Teacher Prep
Required Preparation
Acquire devices that can run Desmos (recommended) or other graphing technology. It is ideal if students each have their own device. (Desmos is available under Math Tools.)
Narrative
In this Warm-up, experiment with the graphical effects of multiplying by a factor both sides of an equation in two variables.
Students are prompted to multiply one equation in a system by several factors to generate several equivalent equations. They then graph these equations on the same coordinate plane that shows the graphs of the original system. Students notice that no new graphs appear on the coordinate plane and reason about why this might be the case.
The work here reminds students that equations that are equivalent have all the same solutions, so their graphs are also identical. Later, students will rely on this insight to explain why we can multiply one equation in a system by a factor—which produces an equivalent equation—and solve a new system containing that equation instead.
Launch
Display the equation x+5=11 for all to see. Ask students:
- "What is the solution to this equation?" (x=6)
- "If we multiply both sides of the equation by a factor, say, 4, what equation would we have?" (4x+20=44) "What is the solution to this equation?" (x=6)
- "What if we multiply both sides by 100?" (100x+500=1,100) "By 0.5?" (0.5x+2.5=5.5)
- "Is the solution to each of these equations still x=6?" (Yes)
Remind students that these equations are one-variable equations, and that multiplying both sides of a one-variable equation by the same factor produces an equivalent equation with the same solution.
Ask students: "What if we multiply both sides of a two-variable equation by the same factor? Would the resulting equation have the same solutions as the original equation?" Tell students that they will now investigate this question by graphing.
Arrange students in groups of 2–4, and provide access to graphing technology to each group. To save time, consider asking group members to divide up the tasks. (For example, one person could be in charge of graphing while the others write equivalent equations, and all analyze the graphs together.)
Student Task
Consider two equations in a system:
\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}
- Use graphing technology to graph the equations. Then, identify the coordinates of the solution.
-
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 21. Let’s call the resulting equations A1, A2, and A3. Record your equations here:
- Equation A1:
- Equation A2:
- Equation A3:
- Graph the equations that you generated. Make a couple of observations about the graphs.
Sample Response
- Sample response:
- Equation A1: 8x+2y=2
- Equation A2: -20x−5y=-5, or 20x+5y=5
- Equation A3: 2x+21y=21
- Sample response: No new graphs appeared. The graphs of the equivalent equations are identical to that of Equation A. The graphs of Equations A1, A2, and A3 all intersect the graph of Equation B at (1,-5).
Activity Synthesis (Teacher Notes)
Invite students to share their observations about the graphs they created. Ask students why the graphs of Equations A1, A2, and A3 all coincide with the graphs of the original Equation A. Discuss with students:
- "How can we explain the identical graphs?" (Equations A1, A2, and A3 are equivalent to Equation A, so they all have the same solutions as Equation A, and their graphs are the same line as the graph of Equation A.)
- "What move was made to generate Equations A1, A2, and A3? Why did it create equations that are equivalent to Equation A?" (The two sides of Equation A were multiplied by the same factor, which keeps the two sides equal.)
To further illustrate that Equations A1, A2, and A3 are equivalent to Equation A, and if time permits, consider:
- Using tables to visualize the identical (x,y) pairs. For example:
4x+y=1
x y 0 1 1 -3 2 -7 3 -11 20x+5y=5
x y 0 1 1 -3 2 -7 3 -11 2x+21y=21
x y 0 1 1 -3 2 -7 3 -11 - Remind students that, earlier in the unit, they saw that isolating one variable is a way to see if two equations are equivalent. If we isolate y in Equations A, A1, A2, and A3, the rearranged equations will be identical: y=-4x+1.
Standards
- A-REI.1·Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
- A-REI.1·Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
- A-REI.1·Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
- HSA-REI.A.1·Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method.
- A-REI.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
- A-REI.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
- A-REI.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
- HSA-REI.C.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.