Unit 8A: Systems of Equations/Inequalities
In this unit, you will revisit many of the functions studied in units 1-5 as well as the conics from unit 6. However,
instead of looking at one equation at a time, there will be two or more equations and our goal is to find a common
solution(s). Let’s refresh vocabulary and possible solution sets using basic linear systems.
1A.
⎧2x − y = 5 ⎫
⎨
⎬
⎩ x + 4y = 7 ⎭
2A.
⎧2y = 10 + x ⎫
⎨
⎬
⎩6y + 18 = 3x ⎭
3A.
⎧2x − y = −2 ⎫
⎨
⎬
⎩ y = 2 + 2x ⎭
1B.
2B.
3B.
1C. Number of solutions:
2C. Number of solutions:
3C. Number of solutions:
1D. Solution:
2D. Solution:
3D. Solution:
The following techniques can be used for solving systems in this unit. Create (but don’t solve) systems that would
be best solved by each technique.
•
Graphing:
•
Substitution:
•
Elimination:
EXAMPLES: Solve each system by graphing, substitution, or elimination.
4.
⎧2x + y = 1 ⎫
⎨
⎬
⎩ 3x + 4y = 14 ⎭
Describe system:
Possible # solutions:
6.
⎧x2 − y = 2 ⎫
⎨
⎬
⎩2x − y = −1⎭
Describe system:
Possible # solutions:
⎧ x − 2y + z = 1 ⎫
⎪
⎪
5. ⎨ y + 2z = 5
⎬
⎪ x + y + 3z = 8 ⎪
⎩
⎭
Describe system:
Possible # solutions:
7.
⎧ x 2 + y 2 = 100 ⎫
⎨
⎬
⎩ 3x − y = 10 ⎭
Describe system:
Possible # solutions:
Systems of Inequalities
When solving a system of inequalities, our only option for solving is to graph since we are looking for region where
the solutions lie. Use different colors to graph each inequality.
1.
⎧ 3x + y ≥ −3⎫
⎨
⎬
⎩ x + 2y ≤ 4 ⎭
⎧ x 2 + y 2 < 25 ⎫
3. ⎨
⎬
⎩ x + 2y ≥ 5 ⎭
⎧ x + 3y ≤ 12 ⎫
⎪x + y ≤ 8 ⎪
⎪
⎪
2. ⎨
⎬
⎪x ≥ 0
⎪
⎪⎩ y ≥ 0
⎪⎭
⎧y ≥ x3
⎫
⎪
⎪
4. ⎨2x + y ≥ 0 ⎬
⎪ y ≤ 2x + 6 ⎪
⎩
⎭
Applications
System of Two Linear Equations: A man invests his savings in two accounts, one paying 6% and the other paying
10% simple interest per year. He puts twice as much in the lower-yielding account because it is less risky. His
annual interest income is $3,520. How much did he invest at each rate?
System of Two Non-Linear Equations: The altitude of an isosceles triangle drawn to its base is 3 centimeters,
and its perimeter is 18 centimeters. Find the length of its base.
System of Inequalities: A publishing company publishes no more than 100 books every year. At least 20 of these
are nonfiction, but the company always publishes at least as many fiction as nonfiction. Find a system of
inequalities that describes the possible numbers of fiction and nonfiction books that the company can produce each
year consistent with these policies. Graph the solution set.
PRACTICE PROBLEMS
Solve each system of equations using methods of your choice.
1.
⎧x − y = 2 ⎫
⎨
⎬
⎩2x + 3y = 9 ⎭
Describe system:
Possible # solutions:
3.
⎧x − y = 4 ⎫
⎨
⎬
⎩ xy = 12 ⎭
2.
⎧y = x2
⎫
⎨
⎬
⎩ y = x + 12 ⎭
Describe system:
Possible # solutions:
4.
⎧⎪ 3x 2 + 2y = 26 ⎫⎪
⎨ 2
⎬
⎪⎩5x + 7y = 3 ⎪⎭
Describe system:
Possible # solutions:
Describe system:
Possible # solutions:
3 ⎫
⎧
⎪log x + log y = ⎪
5. ⎨
2 ⎬
⎪⎩2 log x − log y = 0 ⎪⎭
⎧ 4x − 3y + z = −8 ⎫
⎪
⎪
6. ⎨−2x + y − 3z = −4 ⎬
⎪ x − y + 2z = 3
⎪
⎩
⎭
Describe system:
Possible # solutions:
Describe system:
Possible # solutions:
Solve each system of inequalities.
⎧x > 0
⎫
⎪y > 0
⎪
⎪
⎪
7. ⎨
⎬
x
+
y
<
10
⎪
⎪
⎪⎩ x 2 + y 2 > 9 ⎪⎭
9. Find the system of inequalities that satisfies the
solution shown here:
8.
⎧x − y > 0 ⎫
⎨
⎬
⎩ 4 + y ≤ 2x ⎭
10. Find the system of inequalities that satisfies the
solution shown here:
11. Brandon recently received an inheritance of $50,000. His friends advised him to invest in at least three
different funds. First, Brandon selects a money-market fund. Then he chooses Dr. Pepper Co. stock because he
loves pop, and Apple Inc. because he also loves his iPhone. A financial advisor estimates that the money market
fund will yield a return of 5% in the next year, the Dr. Pepper stock 9%, and the Apple stock 16%. Brandon wants a
total first-year return of $4,000. To avoid excessive risk, he decides to invest three times as much in the moneymarket fund as in the Apple fund. Set up and solve a system to determine how much to invest in each fund.
⎧y ≥ 0
⎫
⎪y ≤ 5
⎪
⎪
⎪
12. Find the area of the region formed by the system: ⎨
⎬.
x
+
y
≤
25
⎪
⎪
⎪⎩ x ≥ 15
⎪⎭