Download ALGEBRA II 2A.3D Supporting

ALGEBRA II
2A.3D Supporting
2A.3 Systems of equations and inequalities. The student applies mathematical processes to formulate systems of equations and
inequalities, use a variety of methods to solve, and analyze reasonableness of solutions. The student is expected to:
(D) determine the reasonableness of solutions to systems of a linear equation and a quadratic equation in two variables.
2A.3D
What Readiness Standard(s) or concepts from the Readiness Standards does it support?
• 2A.3A formulate systems of equations, including systems consisting of three linear equations in three variables and systems consisting of two equations, the first linear and the
second quadratic
• 2A.4F solve quadratic and square root equations
May be adjusted according to local curriculum
Instructional Implications
In accordance with the standard, students should be able to determine the reasonableness of solutions to systems consisting of a linear and a quadratic equation. In general,
students can determine reasonableness by evaluating the equations, generating tables, and graphing the related functions.
For example, consider the system given by the equations 4x + 2y = 25 and y = -0.5x2 + 3x. Suppose a student estimates that the system has a solution at (5, 2.5). The first strategy
to determine the reasonableness of this solution is to evaluate each equation at the given values of x and y. Here, 4(5) + 2(2.5) = 20 + 5 = 25; so the point checks in the linear
equation. Also, when x = 5, the quadratic equation yields y = -0.5(5)2 + 3(5) = -12.5 + 15 = 2.5. So, since the values make both equations true, the ordered pair is a solution to the
system. However, such systems can have more than one solution. (See 2A.3C.) To determine if other solutions exist, students can generate tables or graphs. This may require
rewriting the linear equation in slope-intercept form, as shown below.
y1 = -2x + 12.5
y2 = -0.5x2 + 3x
x
y1
y2
1
10.5
2.5
2
8.5
4
3
6.5
4.5
4
4.5
4
2.5
5
2.5
6
0.5
0
7
-1.5
-3.5
Note that the tables have only the point (5, 2.5) in common, and the graphs intersect only at this point. Here, these representations confirm that (5, 2.5) is indeed the only solution.
For this system and others like it, reasonableness of solutions can be determined by evaluating the equations and generating tables and graphs.
Rigor Implications
Academic Vocabulary
• System
• Linear equation
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• Quadratic equation
• Solution
• Apply
• Formulate
• Use
50
• Analyze
• Solve
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