options: 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.

Solve systems of linear equations exactly and approximately (e.g., with graphs), focusing on pairs of linear equations in two variables.

Find the inverse of a matrix if it exists and use it to solve systems of linear equations (using technology for matrices of dimension 3 × 3 or greater).

Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. For example, find the points of intersection between the line y = –3x and the circle x^(2 )+ y^(2) = 3.

Represent a system of linear equations as a single matrix equation in a vector variable.
problem_activity: Warm-up
In this activity, students have an opportunity to notice and make use of structure (MP7) in order to identify a point where the graphs of two given functions intersect. The work here leads directly into the next activity in which students use algebraic methods to identify all points of intersection between two polynomials.
Launch
Display one problem at a time. Give students quiet think time for each problem and ask them to give a signal when they have an answer and a strategy. Keep all problems displayed throughout the talk. Follow with a whole-class discussion.
Representation: Internalize Comprehension.
To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory; Organization
Student Facing
Mentally identify a point where the graphs of the two functions intersect, if one exists.
\(f(x)=x\)
and
\(g(x)=3\)
\(j(x)=(x+3)(x-3)\)
and
\(k(x)=0\)
\(m(x)=(x+3)(x-3)\)
and
\(n(x)=(x-3)\)
\(p(x)=(x+5)(x-5)\)
and
\(q(x)=(x+3)(x-3)\)
Student Response
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Activity Synthesis
Ask students to share their strategies for each problem. Record and display their responses for all to see. To involve more students in the conversation, consider asking:
“Who can restate
\(\underline{\hspace{.5in}}\)
’s reasoning in a different way?”
“Did anyone have the same strategy but would explain it differently?”
“Did anyone solve the problem in a different way?”
“Does anyone want to add on to
\(\underline{\hspace{.5in}}\)
’s strategy?”
“Do you agree or disagree? Why?”
If not brought up during the discussion, remind students that these are called systems of equations. Ask, “What are some ways you have solved systems of equations in the past?” (Graphing, substitution, elimination.) Tell students that in the next activity, they are going to think about how to use substitution to solve systems of equations in which at least one of the equations is a quadratic.
Speaking: MLR8 Discussion Supports.
Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse what they will say with a partner before they share with the whole class.
Design Principle(s): Optimize output (for explanation)


correct_option_index: 3