Download Solving Systems of Linear Equations with Two Variables Algebraically

Chapter 6 | Linear Systems
8. By determining the slope and y-intercept of each pair of equations in Problems 4 and 6, classify the system of
linear equations as dependent or independent.
9. By determining the slope and y-intercept of each pair of equations in Problems 3 and 5, classify the system of
linear equations as consistent or inconsistent.
10. By determining the slope and y-intercept of each pair of equations in Problems 4 and 6, classify the system of
linear equations as consistent or inconsistent.
11. Find the value of 'A' for which the lines Ax - 2y - 5 = 0 and 8x - 4y + 3 = 0 are parallel.
12. Find the value of 'A' for which the lines Ax + 3y = 6 and 3x + 2y = 10 are parallel.
13. Find the value of 'B' for which the lines 3x - 2y + 8 = 0 and 6x - By - 3 = 0 are parallel.
14. Find the value of 'B' for which the lines 3x + By = 5 and 2x - 3y = 1 are parallel.
Solve the following systems of linear equations by graphing:
15. 5x + y + 9 = 0 and x - 3y + 5 = 0
16. 3x - 2y +1 = 0 and y +4x - 6 = 0
17. 3x + 2y = -4 and y + 3 x + 2 = 0
2
18. 2x - y = 6 and 6x - 3y = 15
19. 4x - 2y = 6 and -2y + 4x - 8 = 0
20. 2y - x - 6 = 0 and y = 1 x + 3
2
6.4 |
Solving Systems of Linear Equations with
Two Variables Algebraically
Solving systems of linear equations using algebraic methods is the most accurate for the
following reasons:
(i) It eliminates graphing errors.
(ii) It gives the exact answer with systems of equations that have fractions or that have
fractional answers.
There are basically two methods involved in solving systems of linear equations. They are the
substitution method and the elimination method.
The Substitution Method
The substitution method is preferable when the coefficient of any one of the variables is either 1
or -1 in either one of the equations in the system.
In this method, the following steps are used to solve systems of two linear equations with two variables:
1. Solve one of the equations for either x or y.
2. Substitute that result into the second equation to obtain an equation with one variable.
3. Solve the equation obtained from the step above for that variable.
4. Substitute that value into any one of the equations to find the value of the other variable.
Example 6.4(a)
Solving a System of Linear Equations by Substituting for the Variable 'y'
Solve this system:
y - 3x + 2 = 0
2y + x - 10 = 0 Equation (1)
Equation (2)
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Chapter 6 | Linear systems
Solution
From Equation (1), y - 3x + 2 = 0
y = 3x - 2
Substituting this in Equation (2)
2y + x - 10 = 0
Solving for 'x',2(3x - 2) + x - 10 = 0
6x - 4 + x - 10 = 0
7x = 14
x=2
Substituting x = 2 in Equation (1),
y = 3x - 2 = 3(2) - 2 = 4
Therefore, the solution is (2, 4).
Example 6.4(b)
Solving a System of Linear Equations by Substituting for the Variable 'x'
Solve this system:
x + 2y = 6
Equation (1)
4x + 3y = 4 Equation (2)
Solution
x = 6 - 2y
From Equation (1), Substituting in Equation (2), 4(6 - 2y) + 3y = 4
Solving for 'y',
24 - 8y + 3y = 4
20 = 5y
y=4
Substituting y = 4 in equation (1),
x = 6 - 2y = 6 - 2(4) = -2
Therefore, the solution is (-2, 4).
The Elimination Method
The elimination method is preferable when none of the the variables' coefficients are either
1 or -1, in either one of the equations in the system.
In this method, the following steps are used to solve systems of two linear equations with two
variables:
1. Write both equations in the form Ax + By = C.
2. Simplify the equations to eliminate decimals and fractions.
3. Choose a variable to eliminate.
4. Multiply one or both equations by a constant to obtain the least common multiple for the
coefficient of the variable to be eliminated.
5. Add or subtract the two equations so that the variable is eliminated.
6. Solve the equation for the remaining variable.
7. Substitute that value into any one of the equations to find the value of the other variable.
Chapter 6 | Linear Systems
Example 6.4(c)
Solving System of Equations by Eliminating the Variable 'x'
Solve the system:
Solution
2x + 3y = 13
Equation (1)
-3x + 6y = 12 Equation (2)
Choose the variable 'x' to eliminate. It has coefficients of 2 and -3.
Multiplying Equation (1) by 3, and multiplying Equation (2) by 2,
3(2x + 3y) = 3(13) 2(-3x + 6y) = 2(12)
6x + 9y = 39
Equation (3)
-6x + 12y = 24
Adding Equations (3) and (4), we obtain,
(6x + 9y) + (-6x + 12y) = 39 + 24
6x - 6x + 9y + 12y = 63
21y = 63,
63
y=
=3
21
Substituting y = 3 in Equation (1), we obtain,
2x + 3y = 13
Solving for 'x',2x + 3(3) = 13
2x = 13 - 9 = 4
x=2
Therefore, the solution is (2, 3).
Example 6.4(d)
Solving Systems of Equation by Eliminating the Variable 'y'
Solve the system:
3x + 2y = 8
Equation (1)
8x + 5y = 18 Solution
Equation (2)
Choose the variable 'y' to eliminate.
It has coefficients of 2 and 5.
Multiplying Equation (1) by 5, we obtain,
15x + 10y = 40
Equation (3)
Multiplying Equation (2) by 2, we obtain,
16x + 10y = 36
Subtracting Equation (3) from (4), we obtain,
x = -4
Substituting x = -4 into Equation (1), we obtain,
3(-4) + 2y = 8
Rearranging,2y = 20
y =10
Therefore, the solution is (-4, 10).
Equation (4)
Equation (4)
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Chapter 6 | Linear systems
Example 6.4(e)
Solving Systems of Equations by Elimination
Anna invested a total of $10,000 in funds A and B. The amount in fund A earned an interest of 4%
per annum and the amount in fund B earned an interest of 5% per annum. If the total interest earned
from both these funds at the end of one year was $470, how much did she invest in each fund?
Solution
Let the amount invested in Fund A be $A, and the amount invested in fund B be $B.
A + B = 10,000
Also,
Equation (1)
4%(A) + 5%(B) = $470
0.04A + 0.05B = 470
Equation (2)
Choose the variable 'B' to eliminate.
It has coefficients of 1 and 0.05.
Multiplying Equation (1) by 0.05, we obtain,
0.05A + 0.05B = 500
Equation (3)
Substracting Equation (2) from Equation (3),
(0.05A + 0.05B) - (0.04A + 0.05B) = 500 - 470
0.01A = 30
A = 3000
Substituting A = 3000 in Equation (1), we obtain,
3000 + B = 10,000
B = 7000
Therefore, she invested $3000 in Fund A and $7000 in Fund B.
6.4 |
Exercises Answers to the odd-numbered problems are available at the end of the textbook
Solve the following system of equations by using the substitution method:
1. a. y - 3x + 8 = 0 and y - x + 4 = 0
b. 3x - 2y = 8 and x + 3y = 15
2. a. 4x - 7y + 6 = 0 and x - 3y + 2 = 0
b. 6x - 9y + 2 = 0 and x - 2y - 5 = 0
3. a. 5x + 4y = 14 and x - 3y = -1
b. 3x - 2y = 20 and y + 4x = 23
4. a. x - 9y = 6 and 3x - 7y =16
b. 9x - 2y = 12 and 5y + 3x = 21
5. a. x - 3y = 12 and 5x + 2y = 9
b. 7x - 5y + 3 = 0 and -3x + y + 1 = 0
6. a. 3x + y = -8 and 3x - 4y = -23
b. 6x - 5y + 13 = 0 and -3x + y - 8 = 0
Solve the following system of equations by using the elimination method:
7. a. 3y + 2x = 24 and 2x - 2y = 14
b. 5x + 3y = 19 and 3x - 5y = -9
8. a. 7x - 3y = -5 and 5x - 9y = 7
b. 5x - 3y = 2 and 3x - 5y = 7
9. a. 5x - 7y = 19 and 2x + 3y = -4
b. 2y + 3x = 14 and 9x - 4y = 2
Chapter 6 | Linear Systems
10. a. 9x + 8y = 10 and 3x + 2y = 4
b. 3x - 5y = 4 and 5x + 3y = -16
11. a. 5x - 2y + 1 = 0 and 2x - 3y - 4 = 0
b. 3y + 7x = 15 and 3x + 5y +1 = 0
12. a. 4x + 5y = 11 and 2x + 3y = 5
b. 9y + 4x - 1 = 0 and 4x + 5y +3 = 0
Solve the following system of equations:
13. 0.5x - 0.3y = -1.2 and 0.2x - 0.7y = 0.1
14. 1.2x + 0.6y = 0 and 3.5x + 1.7y = 0.01
15. 0.4x - 0.5y = -0.8 and 0.3x - 0.2y = 0.1
16. 0.2x - 0.3y = -0.6 and 0.5x + 0.2y = 2.3
17. 0.7x - 0.4y = 2.9 and 0.6x - 0.3y = 2.4
18. 1.5x + y = 1 and 0.8x + 0.7y = 1.2
Solve the following system of equations:
y
y
x
x
19. 5 + 6 = 3 and 2 - 3 = 3
21.
5x - 5y = - 5 and x - y = 2
3
4
3
2
y
y
x
2x
3
1
23. 4 + 2 = 2 4 and 3 + 6 = 2
20.
22.
24.
x
-
y
y
=- 3 = 2 and x 2
3
3
4 12
x
+
y
2y
= 2 and x +
= 4
2
6
3
3
6
4
3x + y = 1 and x + y = 1
10
3
3
2
5
2
25. 4(x - 3) + 5(y + 1) = 12 and (y + 7) - 3(x + 2) = 1
26. 4(x + 3) - 3(y + 4) = 21 and 2(x + 4) + 5(y - 3) = 10
27. 2(x - 2) - 3(y - 1) = 11 and 5(x + 1) + 2(y - 4) = 8
28. 3(x + 1) - 6(y + 2) = 6 and 5(2x - 4) + 7(y + 1) = -17
29. 3(2x + 1) - 2(y + 7) = -1 and 4(x + 5) + 3(y - 1) = 28
30. 2(3x + 2) + 5(2y + 7) = 13 and 3(x + 1) - 4(y - 1) = -15
31. Meals for two adults and three children cost $48, whereas, meals for three adults and two children cost $52. Find
the cost of the meal for one adult.
32. Three DVDs and four movie tickets cost $94.00, whereas, four DVDs and three movie tickets cost $81. How much
did one DVD cost?
33. The sum of the ages of a son and his father in years was 92. The difference in their ages was 28. How old were
the son and father?
34. The sum of two numbers was 56 and their difference was 22. What were the numbers?
35. Benjamin invested a portion of $25,000 at 3% per annum and the remainder at 4% per annum. The total interest
on the investment for the first year was $900. How much did he invest at each rate?
36. Naomi invested a portion of $10,000 at 4% per annum and the remainder at 2.5% per annum. The total interest on
the investment for the first year was $370. How much did she invest at each rate?
37. Company A charged a one-time setup fee of $40 and $1.25 per page to print a book. Company B charged a one-time
setup fee of $25 and $1.30 per page to print a book.
a. Where would it have been cheaper to print a book that has 160 pages?
b. How many pages should a book have for it to have the same printing cost at either company?
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Chapter 6 | Linear systems
38. Arthur’s Electric Company charged $75 for a service call and an additional $45 per hour to repair a furnace.
Vladimir’s Furnaces Inc. charged $40 for a service call and an additional $50 per hour to repair the same furnace.
a. Where would it have been cheaper to repair a furnace if it took 8 hours for the repair job?
b. At how many hours will the cost be the same to repair the furnace at either company?
39. 450 tickets were sold at an opera in Calgary. Tickets for adults were sold for $9.50 each and tickets for children were
sold for $5.50 each. If a total of $3675 was collected from ticket sales, how many tickets of each kind were sold?
40. Maggie had 50 bills consisting of $20 bills and $5 bills. The total dollar value of the bills was $310. How many of
each type of bill did she have?
|
6 Review Exercises
Answers to the odd-numbered problems are available at the end of the textbook
1. In which quadrant or axis do the following points
lie?
a. A (5, -1)
b. B (-2, 3)
c. C (3, 0)
d. D (4, -2)
e. E (2, 0)
f. F (0, 4)
2. In which quadrant or axis do the following points
lie?
a. A (-4, 5)
b. B (-5, 0)
c. C (-2, -7)
d. D (0, -3)
e. E (6, 6)
f. F (5, 4)
3. Plot the following points and join them in the order
A, B, C, D. Identify the type of quadrilateral and
find its area and perimeter.
a. A (6, -3)
b. B (6, -6)
c. C (-2, -6)
d. D (-2, -3)
4. Plot the following points and join them in the order
P, Q, R, S. Identify the type of quadrilateral and
find its area and perimeter.
a. P (-2, 4)
b. Q (-8, 4)
c. R (-8, -2)
d. S (-2, -2)
5. Graph the following equations using a table of
values with four points:
a. 4x - y = 2 b. x + y - 4 = 0
c. y = 1 x +
+22
2
6. Graph the following equations using a table of
values with four points:
a. 2x + 3y = 12 b. x + 2y - 4 = 0
c. y = - 1 x - 2
3
7. Graph the following equations using the x-intercept,
y-intercept, and another point.
a. 3x - 4y = 12
b. x - 2y - 6 = 0
c. y = 4x
8. Graph the following equations using the x-intercept,
y-intercept, and another point.
a. x - 2y = -1 b. 3x + y - 4 = 0
c. x = 2y
9. Graph the following equations using the slope and
y-intercept method.
a. y = 4x + 6 b. y = - 3 x - 1
4
c. 3x + 2y - 12 = 0
10. Graph the following equations using the slope and
y-intercept method.
a. y = 5x + 4 b. y = - 1 x - 1
3
c. 2x + 3y + 6 = 0
11. Find the equation of the line that passes through
the following points:
a. (3, 2) and (7, 5) b. (5, -4) and (-1, 4)
c. (1, -2) and (4, 7)