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Systems of Equations
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Full Set
Which value is a solution to the equation
8
-2
3x + 4 = 19 ?
5
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Systems of Equations
1. Which ordered pair(s) is a solution to the system
3x + 2y = 14
(1,2)
(-3,8)
and
5x - 2y = 2
(2,4)
(0,5)
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2. Which ordered pair(s) is a solution to the system
and
(-1,5)
(7,-2)
(4,3)
(-1,-2)
1. Solve this system by graphing (on paper)
y = 2/3 x + 1
and
y = -1/2 x + 8
2
2. Solve this system by graphing (on paper)
3x - 2y = 12
and
6x + 2y = 6
3. Solve this system using the calculator and the graphical method. Round to the nearest
tenth.
2x - 5y = 8
and
5x + y = -2
4. Solve this system using the calculator and the graphical method. Round to the
nearest tenth.
y = -4/5 x + 3
and
3x - 2y = 10
3
5. Solve this system using the calculator and the graphical method. Round to the nearest
tenth.
3.12x + 4.65y = 28.89
and
1.89x - 2.35y = 17.67
The Matrix Method for Solving a Linear System
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Solving a System by the Method of Substitution
How do you know when to use this algebraic method? ANSWER: One of the
variables , either x or y, is alone and easily isolated.
Steps:
• isolate the "easy" variable
• substitute into the other equation
• solve for the second variable
• substitute the now-known variable into either
equation to find the remaining variable
1. 3x + y = 3
2x - 5y = 19
2. 5x - 2y = -12
x + 3y = 1
Steps:
• isolate the "easy" variable
• substitute into the other equation
• solve for the second variable
• substitute the now-known variable into either
equation to find the remaining variable
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3.
Steps:
• isolate the "easy" variable
• substitute into the other equation
• solve for the second variable
• substitute the now-known variable into either
equation to find the remaining variable
4.
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5.
The Elimination Method
1. Solve 3x + 2y = 12
4x - 3y = -1
This method can only be used with linear systems.
The Steps: • determine the variable you want to eliminate
• multiple each equation by values so that the
coefficients of that variable are equal
• add or subtract the two equations to
eliminate the variable
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2. Solve
5x - 4y = 14
3x + 2y = 4
3. Solve
6x - 5y = -27
5x + 2y = -4
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4. Solve
Prepare these system for solving by the matrix method.
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Review: determining the number of solutions to a quadratic equation
Determining the Number of Solutions to a Linear System of Equations
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Determine the number of solutions of each linear system.
1. 2x - 3y = 3
8x - 4y = -3
2. 2x - y = 1
4x - 2y = 6
3. 4x + 2y = 6
6x + 3y = 9
Problem Solving - Part A
1. The perimeter of a rectangle is 88 m while the area is 420 sq. m. Find the
dimensions of the rectangle.
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2. The amount of fence needed for the field sketched below is 83 m. The area
of the field is 275 sq. m. Find the dimensions of the field.
3. The diagonal of a rectangle has a length of 5 m. The area is 12 squ. m. Find the
dimensions of the rectangle.
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4. When a skydiver is free-falling his height above ground is given by the formula
When the skydiver is falling using the parachute the height above ground is given
by
a) How many seconds until the parachute gets opened?
b) How high above ground was the parachute opened?
5. Given the polynomial function :
When it is divided by x + 1 the remainder is 8 and when it is divided by x - 2 the
remainder is 14. Find the values of m and n.
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Problem Solving - Part B
1. The sum of two numbers is 26 and their product is 126. what are the numbers?
2. Three balls and two bats cost $90. Two balls and seven bats cost $145. Find
the price of each ball and each bat.
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3. Membership in a club requires payment of an initiation fee plus a monthly fee.
Seven months cost $95 and 12 months cost $144. Find the initiation and monthly fees.
4. Bolts are kept in a metal box. The total mass is 3170 g for 60 bolts and 1700 g for
25 bolts. Calculate the mass of the box and each bolt.
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5. A total of $1600 was invested into two businesses. The first gained 21% while
the second lost 8% over the year. The net gain was $75. Calculate the amount
invested in each business.
6. Five litres of white milk and eight litres of chocolate milk cost $14.10. White
milk is going up in price by 10% while chocolate milk is going up 50%. The same
amount of both now costs $19.35. Find the original price of each.
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Linear Systems - 3 equations with 3 unknowns
Here is a general procedure that works for all systems of 3 equations. Individual
problems may have shorter solutions.
steps:
• pick a pair equations and eliminate one of the variables
• pick another pair of equations and eliminate the same variable
• solve the resulting system of two equations in two unknowns
1. 2x + 3y - 2z = 10
2. 5x - y + 2z = 13
and
and
3x + 2y + 2z = 5
2x + 3y - 5z = 27
and
and
x + y - 3z = 6
x - 2y - 2z = 2
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3. x = 1 + 2y
and
x+z=8
and
y-z=2
Systems of Linear Inequalities
1. Sketch the graph of
2x - 5y < 10
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2. Sketch the graph of 3x + 4y - 12 < 0 and x >=1
3. Sketch the graph of 3x - 5y > -15 and x + 2y < 8
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5. Sketch the graph of
y > 0 and x > -2 and 5x + 3y < 15
Review
1. Which point(s) is a solution to the system :
3x - 7y + 5 = 18
(3,5)
and
(2,-1)
y = 2x - 5
(4,-1)
(-2,3)
2. Which point(s) is a solution to the system:
3x + 7y -3 > 0
(3,2)
and
(-7,8)
5x - 2y < -6
(-5,-4)
(-2,5)
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3. Solve by graphing.
3x - 5y = -15
and
2y + 3y = 6
Check two ways with the graphing calculator.
4. Solve by elimination:
3x - 2y = 17
and
2x + 3y = 7
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5. Solve by substitution:
6. Solve by elimination:
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7. Solve -x + y + z = -2
x-y+z=0
x+y-z=4
algebraically.
8. Two towns are 300 km apart. It takes 7.5 hours by boat to travel this
distance against the current and only 6 hours with the current. Calculate the
speed of the current and the boat in calm water.
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9. Science books cost twice as much as math books. Ten math books and five science
books cost $900. Find the price of each book.
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