Advanced Intermediate Algebra - Chapter 3 Summary
SOLVING SYSTEMS OF LINEAR EQUATIONS
1. Your job is to distribute all 255 apples among the horses and pigs so that each pig gets 3 apples and each
horse gets 5 apples. The number of horses and pigs total 65 animals. How many of the animals are pigs and
how many are horses?
2. At a park, children ride a train for $1, adults pay $2, and senior citizens $1. On a given day, 100 passengers
paid a total of $160. There were 10 more children than senior citizens. Find the number of children riders.
Advanced Intermediate Algebra – Ch. 3_Summary
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3. What does it mean to solve a system of two equations in two variables?
If a system of equations has at least one solution, it is ___________________________.
If a system of equations has exactly one solution, it is ___________________________.
If a system of equations has an infinite number of solutions, it is _________________________.
METHODS FOR SOLVING A SYSTEM OF EQUATION:
SOLVE BY GRAPHING
4) 6x – 2y = 7
2x + y = 4
Graph the equations on the same coordinate plane.
The point of intersection represents the solution.
Check the solution by substituting the coordinates into
each equation.
Solution: ______________
Classify this system of equations:
SOLVE BY SUBSTITUTION
Advanced Intermediate Algebra – Ch. 3_Summary
SOLVE BY ELIMINATION
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SOLVE BY GRAPHING
5) 3x – 2y = 6
6x – 4y = 12
Solution: ______________
Classify this system of equations:
SOLVE BY SUBSTITUTION
SOLVE BY ELIMINATION
SOLVE BY GRAPHING
x y
6)
+ =1
4 3
3
y = - x +1
4
Solution: ______________
Classify this system of equations:
SOLVE BY SUBSTITUTION
Advanced Intermediate Algebra – Ch. 3_Summary
SOLVE BY ELIMINATION
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7. Solve the system of equations.
x–y+z=0
2x – 3z = -1
-x – y + z = 4
9) Solve the system of equations.
2A + B = 4
B–C=5
2A + 2B – C = 9
Advanced Intermediate Algebra – Ch. 3_Summary
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10. TRAFFIC CONTROL
The given figure shows the intersection of three oneway streets. To keep traffic moving, the number of
cars per minute entering an intersection must equal the
number of cars exiting that intersection.
For intersection I1, x + 10 cars enter and y + 14 exit per
minute. Thus x + 10 = y + 14
a) Write an equation for intersection I2 that keeps
traffic moving.
b) Write an equation for intersection I3 that keeps
traffic moving.
c) Solve the system of equations for x, y, and z.
d) If construction limits z to 4 cars/minute, how many
cars per minute must pass between the other
intersections to keep traffic moving?
Advanced Intermediate Algebra – Ch. 3_Summary
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11. Find the quadratic function y = ax2 + bx + c to model the following data:
X (Age of
Y (Average number of
driver)
Automobile Accidents
per Day in the US)
25
300
35
100
55
250
Advanced Intermediate Algebra – Ch. 3_Summary
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12. True or false: A system of equations has an infinite number of solutions if there are more variables than
equations. Justify your answer.
ìax + by = c
13. State the conditions (placed on the coefficients a, b, c, d, e, and f ) for which the system í
is
îdx + ey = f
a) consistent and independent (i.e. has only one solution)
b) consistent and dependent (i.e. has an infinite number of solutions)
c) inconsistent (has no solution)
Advanced Intermediate Algebra – Ch. 3_Summary
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SOLVING SYSTEMS OF LINEAR INEQUALITIES BY GRAPHING
1. The available parking area of a parking lot is 600 square meters. A car requires 6 square meters of space,
and a bus requires 30 square meters of space. The attendant can handle no more than 60 vehicles. Set up
and graph a system of linear inequalities to show all possible combinations of cars and buses that meet the
constraints.
2. Cameron has been sent to the store to purchase donuts and juice boxes for the math team. He can spend at
most $50. A dozen of donuts costs $6. A pack of 8 juice boxes costs $5. He needs to buy at least 30 donuts
and 30 juice boxes. Set up and graph the region that shows how many dozens of donuts and packages of
juice boxes that he can purchase. Give an example of three different purchases he can make.
Advanced Intermediate Algebra – Ch. 3_Summary
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SOLVE BY GRAPHING
1. y > -2x + 4
y≤x–2
2. y > x + 1
|x| ≤ 3
3. x – 2y < – 2
1
y < x-3
2
Advanced Intermediate Algebra – Ch. 3_Summary
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4. Find the coordinates of the vertices formed by
the system of inequalities. Find the area of the
region.
x + y ≥ -1
x–y≤6
y≤4
5. Find the area of the region defined by |x| + |y| ≤ 5 and |x| + |y| ≥ 2.
Advanced Intermediate Algebra – Ch. 3_Summary
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LINEAR PROGRAMMING
1. I am thinking of a point with coordinates (x, y) in the coordinate plane that makes the quantity 3x + y
as large as possible. The ordered pair has to meet all of the following conditions:
y≥1
x≤6
y ≤ 2x + 1
What choice(s) of (x, y) would
work?
Advanced Intermediate Algebra – Ch. 3_Summary
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2. A television manufacturer makes console and wide-screen televisions. It is bound by the following
constraints:
• Equipment in the factory allows for making at most 450 console televisions and at most 200 wide-screen
televisions in one month.
• The cost to the manufacturer per unit is $600 for the console TVs and $900 for the wide-screen TVs.
Total monthly costs cannot exceed $360,000.
The profit per unit is $125 for the console TVs and $200 for the wide-screen TVs. How many of each type
of televisions should be made to maximize the profit?
Advanced Intermediate Algebra – Ch. 3_Summary
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3. A school is preparing lunch menus containing foods A and B. The specifications for the two foods are
given in the following table:
Food
A
B
Units of fat per
ounce
1
1
Units of Carbohydrates per
ounce
2
1
Units of protein per ounce
1
1
Each lunch must provide at least 6 units of fat per serving, no more than 7 units of protein, and at least 10
units of carbohydrates. The school can purchase food A for $0.12 per ounce and food B for $0.08 per
ounce. How many ounces of each food should a serving contain to meet the dietary requirements at the
least cost?
Advanced Intermediate Algebra – Ch. 3_Summary
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