A.REI.7: Solve Quadratic-Linear Systems Algebraically and by Graphing
A.REI.7: Solve Quadratic-Linear Systems Algebraically and
by Graphing
Solve systems of equations.
7. 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 x2 + y2 = 3.
Overview of Lesson
- activate prior knowledge
- present vocabulary and/or big ideas associated with the lesson
- connect assessment practices with curriculum
- model an assessment problem and solution strategy
- facilitate guided practice of student activity
Optional: Provide or allow students to create additional problem sets
- facilitate a summary and share out of student work
Optional HW - Write the math assignment.
Big Idea
Two or more equations together form a system of equations. Any solution common to all equations is a
solution of the system. When one or more of the equations is a quadratic, the system is called a nonlinear system of equations. Non-linear systems of equations are solved in the same way as linear
systems, except there can be more than one solution.
Sample Regents Problem
Solve the following system of equations algebraically.
y  x2  4x  2
y  2x  1
One Solution:
Since both equations have y isolated, the two polynomial expressions can be set equal to
one another.
Substitute the values of x into the simpler question to find the corresponding values of y.
When
,
When
,
y  2x  1
y  2x  1
y  2( 3)  1
y  2(1)  1
y  6  1
y  2 1
y  5
y3
The solutions are (-3,-5) and (1,3)
Check by inputting the two original equations into a graphing calculator and visually
inspect the points of intersection.
REGENTS PROBLEMS
1. Let
and
. On the set of axes below, draw the graphs of
Using this graph, determine and state all values of x for which
and
.
.
2. A company is considering building a manufacturing plant. They determine the weekly production cost at site A to
be
while the production cost at site B is
, where x represents the number of products, in
hundreds, and
and
are the production costs, in hundreds of dollars. Graph the production cost
functions on the set of axes below and label them site A and site B.
State the positive value(s) of x for which the production costs at the two sites are equal. Explain how you
determined your answer. If the company plans on manufacturing 200 products per week, which site should they
use? Justify your answer.
A.REI.7: Solve Quadratic-Linear Systems Algebraically and by Graphing
Answer Section
1. ANS:
b)
a)
when
and
.
Strategy: Input both functions into a graphing calculator and use the table and graph views to construct the graph
on paper and to answer the question.
PTS: 4
2. ANS:
REF: 081435a1
NAT: A.REI.7
TOP: Quadratic-Linear Systems
a)
b) The graphs of the production costs are equal when
.
c) The company should use Site A, because the costs of Site A are lower when
.
Strategy: Input both functions into a graphing calculator and use the table and graph views to construct the graph
on paper and to answer the question.
PTS: 6
REF: 061437a1
NAT: A.REI.7
TOP: Quadratic-Linear Systems