Download Solving Systems of Linear Equations with Two Variables Algebraically

Chapter 6 | Linear Systems
6.3 _
1.
Exercises
Solve the system of equations by graphing.
a. y = 3x - 2 and y = -7x + 8
2.
3.
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b. y = -x + 2 and 2y = -2x + 6
c. y = -4x + 7 and 2y + 8x = 14
Solve the system of equations by graphing.
a. y = 3x + 9 and y = x - 4
b. y = 2x + 6 and 2y = -3x + 6
c. y = -2x + 3 and 2y + 4x = 6
By determining the slope and y-intercept of each pair of equations in Problem 1, classify the system of equations
as dependent or independent.
4.
By determining the slope and y-intercept of each pair of equations in Problem 2, classify the system of equations
as dependent or independent.
5.
By determining the slope and y-intercept of each pair of equations in Problem 1, classify the system of equations
as consistent or inconsistent.
6.
By determining the slope and y-intercept of each pair of equations in Problem 2, classify the system of equations
as consistent or inconsistent.
7.
Find the value of 'A' for which the lines Ax - 2y - 5 = 0 and 8x - 4y + 3 = 0 are parallel.
8.
Find the value of 'B' for which the lines 3x - 2y + 8 = 0 and 6x - By - 3 = 0 are parallel.
9.
Solve the following system of equations by graphing
b. 3x + 2y = -4 and y + 3 x + 2 = 0
a. 5x + y + 9 = 0 and x - 3y + 5 = 0
2
c. 4x - 2y = 6 and -2y + 4x - 8 = 0
10. Solve the following system of equations by graphing
a. 3x - 2y +1 = 0 and y +4x - 6 = 0
c. 2y - x - 6 = 0 and y = 1 x + 3
2
6.4 _
b. 2x - y = 6 and 6x - 3y = 15
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: It eliminates graphing errors.
(i)
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:
190
Chapter 6 | Linear systems
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 Equations by Substituting for the Variable 'y'
Solve this system:
Solution
y - 3x + 2 = 0
Equation (1)
2y + x - 10 = 0
Equation (2)
y - 3x + 2 = 0
From Equation (1),
y = 3x - 2
Substitute this into 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 System of Equations by Substituting for the Variable 'x'
Solve this system:
x + 2y = 6
Equation (1)
4x + 3y = 4 Equation (2)
Solution
From Equation (1),
Substituting in Equation (2),
x=6-2
2yy
4(6 - 2y) + 3y = 4
Solving for 'y',
24 - 8y + 3y = 4
20 = 5y
y=4
Substituting y = 4 in equation (1),
x=6-2
2yy = 6 - 2(4) = -2
Therefore, the solution is (-2, 4).
The Elimination Method
The elimination method is preferable when none of the the variables' coefficient 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:
Chapter 6 | Linear Systems
1. Write both equations in the form Ax + By = C.
2. Simplify the equations to eleminate decimals and fraction.
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.
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)
3y) = 3(13)
6y) = 2(12)
2(-3x + 6y)
6x + 9y
9y = 39
12yy = 24
-6x + 12
(6x + 9
y) + (-6x + 12
y) = 39 + 24
9y)
12y)
Adding Equations (3) and (4)
6x - 6x + 9
9yy + 12y = 63
21y = 63,
63
y=
=3
21
Substituting y = 3 into Equation (1), we obtain,
2x + 3
3yy = 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:
Solution
3x + 2y = 8
Equation (1)
8x + 5y = 18
Equation (2)
Choose the variable 'y' to eliminate.
It has coefficients of 2 and 5.
Multiplying Equation (1) by 5, we obtain,
15x + 10y
10y = 40
Multiplying Equation (2) by 2, we obtain,
Equation (3)
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Chapter 6 | Linear systems
16x + 10y
10y = 36
Solution
continued
Equation (4)
Subtract 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
_ Exercises
Therefore,
the solution is (-4, 10).
6.4
1.
Solve the following by the substitution method:
a. y - 3x + 8 = 0 and y - x + 4 = 0
b. 3x - 2y = 8 and x + 3y = 15
c.x - 3y = 12 and 5x + 2y = 9
2.
Solve the following by the substitution method:
a. 4x - 7y + 6 = 0 and x - 3y + 2 = 0
b. 6x - 9y + 2 = 0 and x - 2y - 5 = 0
c. 3x + y = -8 and 3x - 4y = -23
3.
Solve the following by the substitution method:
a. 5x + 4y = 14 and x - 3y = -1
b. 3x - 2y = 20 and y + 4x = 23
c. 7x - 5y + 3 = 0 and -3x + y + 1 = 0
4.
Solve the following by the substitution method:
a. x - 9y = 6 and 3x - 7y =16
b. 9x - 2y = 12 and 5y + 3x = 21
c. 6x - 5y + 13 = 0 and -3x + y - 8 = 0
5.
Solve the following by the elimination method:
a. 3y + 2x = 24 and 2x - 2y = 14
b. 5x + 3y = 19 and 3x - 5y = -9
c. 5x - 2y + 1 = 0 and 2x - 3y - 4 = 0
6.
Solve the following by the elimination method:
a. 7x - 3y = -5 and 5x - 9y = 7
b. 5x - 3y = 2 and 3x - 5y = 7
c. 4x + 5y = 11 and 2x + 3y = 5
7.
Solve the following by the elimination method:
a. 5x - 7y = 19 and 2x + 3y = -4
b. 2y + 3x = 14 and 9x - 4y = 2
c. 3y + 7x = 15 and 3x + 5y +1 = 0
8.
Solve the following by the elimination method:
a. 9x + 8y = 10 and 3x + 2y = 4
b. 3x - 5y = 4 and 5x + 3y = -16
c. 9y + 4x - 1 = 0 and 4x + 5y +3 = 0
9.
Solve the following equations:
a. 0.5x - 0.3y = -1.2 and 0.2x - 0.7y = 0.1
c. 0.7x - 0.4y = 2.9 and 0.6x - 0.3y = 2.4
b. 0.4x - 0.5y = -0.8 and 0.3x - 0.2y = 0.1
Chapter 6 | Linear Systems
10. Solve the following equations:
a. 1.2x + 0.6y = 0 and 3.5x + 1.7y = 0.01
b. 0.2x - 0.3y = -0.6 and 0.5x + 0.2y = 2.3
c. 1.5x + y = 1 and 0.8x + 0.7y = 1.2
11. Solve the following equations:
a. x + y = 3 and x - y = 3
2
3
5
6
y
b. 5x - 5y = - 5 and x - = 2
3
4
3
2
y
y
c. x + = 2 41 and 2x + = 3
3
6
2
4 2
12. Solve the following equations:
a. x - y = 2 and x - y = - 3
2
6
3
3
4 12
y
2y
b. x + = 2 and x +
=4
4 2
6
3
3
y
y
c. 3x + = 1 and x + = 1
10
5
2
3
3
2
13. Solve the following equations:
a. 4(x - 3) + 5(y + 1) = 12 and (y + 7) - 3(x + 2) = 1
b. 2(x - 2) - 3(y - 1) = 11 and 5(x + 1) + 2(y - 4) = 8
c. 3(2x + 1) - 2(y + 7) = -1 and 4(x + 5) + 3(y - 1) = 28
14. Solve the following equations:
a. 4(x + 3) - 3(y + 4) = 21 and 2(x + 4) + 5(y - 3) = 10
b. 3(x + 1) - 6(y + 2) = 6 and 5(2x - 4) + 7(y + 1) = -17
c. 2(3x + 2) + 5(2y + 7) = 13 and 3(x + 1) - 4(y - 1) = -15
15. 2 adult meals and 3 children meals cost $48, whereas, 3 adult meals and 2 children meals cost $52. How much is
1 adult meal?
16. 3 DVDs and 4 movie tickets cost $94.00, whereas, 4 DVDs and 3 movie tickets cost $81. How much is 1 DVD?
17. The sum of a son's age and of his father's age in years is 92. The difference in their ages is 28. How old are the son
and father?
18. The sum of 2 numbers is 56 and their difference is 22. What are the numbers?
19. Hanna charges $20 for the first shirt that you purchase at her store. However, for every additional shirt, she charges
$15. Write an equation that shows the relationship between her total revenue (y) and number of shirts sold (x). If
you plot a graph of y vs. x, what is the y-intercept and slope of the line that represents the equation?
20. An online music store charges $2 for the first song that you download. For every additional song you download,
you will be charged only $1.50. Write an equation that shows the relationship between the total revenue (y) and the
number of songs sold (x). If you plot a graph of y vs. x, what is the y-intercept and slope of the line that represents
the equation?
21. Henry works in a computer store and earns $500 a month plus a commission of 10% on the sales he makes. The
relationship between his earnings (y) in a month and the number of computers he sells (x) is given by the equation
y = 500 + 0.10x.
a. What would his commission be if his earnings are $14,000 in a month?
b. What would his earnings be if his sales are $250,000 in a month?
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