Math 2
Lesson 7-3: Nonlinear Systems of Equations
Name_______________________________
Date ___________________________
Learning Goals:
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I can use substitution to solve a system of equations in which one equation is linear and one
equation is quadratic.
I can use substitution to solve a system of equations in which one equation is linear and one
equation is an inverse variation.
I can combine different parent functions (adding, subtracting, multiplying, and/or dividing) to write
a function that describes a real-world problem.
In the previous lessons, we focused on solving quadratic equations – square-rooting both sides of the
equation; factoring; completing the square; and the quadratic formula. In this activity, we will further
develop our skills in solving quadratic equations.
Take notes as we solve the following equation:
What does this situation look like graphically? Can we use our calculator to help us check our answer?
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1. Now you try. Solve the following equation BY HAND. When you are finished, enter your smallest
solution into your calculator. Also sketch a graph to illustrate your answer.
x 2  4 x  5  2 x  10
2. When you finish the above problem quickly, try solving below more challenging problem BY HAND.
Check your solution by graphically using your calculator.
When we have a quadratic equation set equal to a linear equation, how many solutions are possible?
Sketch examples of what each type would look like graphically. Also describe what each type will look
like algebraically.
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Can quadratic situations arise in problems that do not have quadratic equations originally?
Take notes as we solve the following equations.
3. Now you try. Solve the following equation BY HAND. Illustrate your solution with a sketch of the
graph. When you are finished, enter your smallest answer into your calculator, then move on to the next
problem.
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4. Solve the following equation for x:
When we have an inverse variation set equal to a linear equation, how many solutions are possible?
Sketch examples of what each type would look like graphically. Also describe what each type will look
like algebraically.
5. Practice the below problems. They are mixed between the two types of problems practiced above. Solve
the equations BY HAND, then check your answer on your calculator.
a.
b.
c.
d.
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The below problems are word problems that will involve solving equations of the same type that you have
practiced above. However, you will have to read the situation and set up the equations. Then solve the
equations BY HAND. Check your solutions using your calculator.
6. Cuyahoga County has $200,000 to spend on salary for summer jobs for student workers.
a.
How many students can be hired if the salary paid to the student workers is $2,000?
b.
If the salary paid to a student worker is represented by p, write a function h  p  that represents the
number of students that can be hired for summer jobs by Cuyahoga County.
Cuyahoga County did some research and found that the function s  p   0.1 p models the number of
students that will apply for a summer job, given the salary is p. Use your equation from part (b) and the
equation for s  p  to write equations or inequalities that model the following situations. In parts (c) and
(d), you do NOT need to solve the equations/inequalities.
c.
For what pay rate will the number of students that can be hired be less than the number of students
that would be interested in work?
d.
For what pay rate will the number of students that can be hired be more than the number of students
that would be interested in work?
e.
Solve for what pay rate will the number of interested workers equals the number that can be hired.
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7. A local theatre is looking at their business plan and finds the following:
8. Based on the relationship between ticket price and number of tickets sold, the show planners figured that
income could be predicted from ticket price as well. They reasoned that since income is equal to the
produce of price per ticket and number of tickets sold, I  p   p  2500  50 p  .
***Put the equation for I  p  in standard form.
9. The estimated operating costs for the theatre are modeled by the equation c  p   22500 100 p . What
price should the theatre set for tickets so that they break even.
10. Use your graphing calculator to help you find an interval when the theatre will make a positive profit.
***Remember, Profit = Income – Expenses.