301
57.
58.
Y
Y
5
6
0
–2
2
X
2
–3
-2
X
0
5
Determine the equations for the following lines:
59. A line parallel to 3y − 2x = 6 and passing through the point P(2, −3).
60. A line parallel to y = 3x − 1 and passing through the point P(−2, −4).
61. A line parallel to 3x − 9y = − 2 and passing through the y-intercept of the line 5x − y = 20.
62. A line parallel to 3x + y = − 2 and passing through the x-intercept of the line 2x + 3y − 4 = 0.
63. A line perpendicular to x + y = 3 and passing through the point P(−2, 5).
64. A line perpendicular to 4x + y + 1 = 0 and passing through the point P(3, 4).
65. A line perpendicular to 2x − y = 5 and passing through the x-intercept of the line 3x + 2y − 6 = 0.
66. A line perpendicular to 3x + y + 9 = 0 and passing through the y-intercept of the line 2x + 3y − 10 = 0.
9.3 Solving Systems of Linear Equations
with Two Variables, Graphically
Introduction
In the previous section, you learned that a linear equation with two variables produces a straight line
when plotted on a graph. If the graph of an equation is linear, then all the points (ordered pairs) on
the line are the solution to that linear equation.
For example, 2x + y = 4 is a linear equation with two variables, x and y. The graph of this equation is a line
and all the points (ordered pairs) on this line are solutions to this equation as shown in the diagram below.
Y
4
All points on the line
are solutions to the
equation 2x + y = 4
3
2
1
-4
-3
-2
-1
0
-1
X
1
2
3
4
Line representing
2x + y = 4
-2
-3
-4
Exhibit 9.3-a Graph of 2x + y = 4
9.3 Solving Systems of Linear Equations with Two Variables, Graphically
302
System of Equations
Two or more equations analyzed together are known as a system of equations. In this section, we will
be analyzing two linear equations with two variables.
The solution to a system of two equations with two variables is an ordered pair of numbers
(coordinates) that satisfy both equations.
If we graph a system of two linear equations, and if they intersect, then the point at which the two
lines intersect will be the solution to both lines.
For example, 2x + y = 4 and x − 2y = −3 form a system of two linear equations. The graphs of these
equations intersect at (1, 2), as shown in the following diagram. This point, (1, 2), is the solution to
the system of two linear equations.
Y
4
Line representing
x - 2y = -3
3
2
This intersecting point
(ordered pair) is the solution to
both equations in the system.
1
-4
-3
-2
0
-1
X
1
2
3
4
-1
Line representing
2x + y = 4
-2
-3
-4
Exhibit 9.3-b Graph of 2x + y = 4 and x − 2y = –3
Graphs of Two Linear Equations May Intersect at One Point, Not Intersect, or Coincide
■■ If they intersect (lines are not parallel), it indicates that there is only one solution.
■■ If they do not intersect (lines are parallel and distinct), it indicates that there is no solution.
■■ If they coincide (lines are the same), it indicates that there are an infinite number of solutions.
­­­­Consistent and Inconsistent Systems
A linear system of two equations that has one or an infinite number of solutions is known as a
consistent linear system.
A linear system of two equations that has no solution is called an inconsistent linear system.
If the graphs of 2 linear
equations intersect at
one point or if the lines
coincide (representing the
same line) then they are
“consistent” as a system.
Otherwise, they are
“inconsistent” as a system.
Do Not Intersect
(Parallel and Distinct)
No Solution
Linear Systems
of Two Equations
Intersect
(Meet at One Point)
One Solution
Coincide
(Same Line)
Infinite Solutions
Chapter 9 | Graphs and Systems of Linear Equations
Inconsistent
System
Consistent
System
303
­­­Dependent and Independent Equations
If a system of equations has an infinite number of solutions then the equations are dependent.
If a system of equations has one or no solution, then the equations are independent.
If the graphs of 2 linear
equations coincide
(representing the same
line), then they are
known as a “dependent
system of equations”,
otherwise they are known
as an “independent
system of equations”.
Do Not Intersect
(Parallel and Distinct)
No Solution
Linear Systems
of Two Equations
Independent
System of Equations
Intersect
(Meet at One Point)
One Solution
Dependent
System of Equations
Coincide
(Same Line)
Infinite Solutions
Intersecting Lines
Slopes of lines:
Different
y-intercepts:
Will be different, unless the lines
intersect on the Y-axis or at the origin.
Number of solutions:
One
Y
May or may not be the same.
ONE SOLUTION
System:
Consistent
Equations:
Independent
X
Exhibit 9.3-c Intersecting Lines
Parallel and Distinct Lines
Slopes of lines:
Same
y-intercepts:
Different
Number of solutions:
None
System:
Inconsistent
Equations:
Independent
Y
NO SOLUTION
X
Exhibit 9.3-d Parallel and Distinct Lines
Coincident Lines
Slopes of lines:
Same
y-intercepts:
Same
Number of solutions:
Infinite
System:
Consistent
Equations:
Dependent
Y
MANY SOLUTIONS
X
Exhibit 9.3-e Coincident Lines
9.3 Solving Systems of Linear Equations with Two Variables, Graphically
304
Solving Linear Systems Graphically
The following steps will help to solve a system of two linear equations with two variables graphically:
Step 1:
Rewrite both equations in the form of y = mx + b ( or Ax + By = C ) and clear off any
fractions or decimal numbers.
Step 2:
Graph the first equation by either using the slope and y-intercept method (or using the
x-intercept and y-intercept) or using a table of values. The graph will be a straight line.
Step 3:
Graph the second equation on the same axes as in Step 2. This will be another straight line.
Step 4:
If the two lines intersect at a point, then the point of intersection is the solution to the given
system of equations, also known as an ordered pair (x, y).
Step 5:
Check the solution obtained by substituting the values for the variables in each of the
original equation. If the answer satisfies the equations, then it is the solution to the given
system of equations.
Note: In Step 2, the order in which the equations are graphed or the method used to graph the equations
does not matter.
Example 9.3-a
Solving and Classifying Systems of Linear Equations
Solve the following system of equations by graphing, and classify the system as consistent or
inconsistent and the equations as dependent or independent.
x – y + 1 = 0
x + y − 3 = 0
Solution
Step 1:
x – y + 1 = 0
Writing the equation in y = mx + b form,
y = x + 1
Equation
x + y −3 = 0
y = –x + 3
Step 2:
Writing the equation in y = mx + b form,
Equation
Equation
y=x+1
:
m = 1 and b = 1
Therefore, the slope is 1 and the y-intercept is (0, 1).
1
Step 3:
: y = –x + 3
Equation
m = –1 and b = 3
Therefore, the slope is – 1 and the y-intercept is (0, 3).
1
and
using the slope and y-intercept form:
Graphing equations
Y
5
4
3
y=x+1
(0, 3)
2
1
-1
Chapter 9 | Graphs and Systems of Linear Equations
(2, 3)
Common Point (1,2)
y = −x + 3
(0, 1)
1
2
3
4
X
305
Step 4:
The two lines intersect at the common point (1, 2).
Step 5:
Check the solution (1, 2) in Equations
, y = x + 1
Equation
and
.
Equation
, y = −x + 3
LS = y = 2
LS = y = 2
RS = x + 1 = 1 + 1 = 2
RS = −x + 3 = −1 + 3 = 2
Therefore, LS = RS
Therefore, LS = RS
Therefore, the solution is (1, 2). The system is consistent (has a solution) and the equations are
independent (lines are not coincident).
Example 9.3-b
Classifying Systems of Linear Equations
Solve this system of equations by graphing, and classify the system as consistent or inconsistent and
the equations as dependent or independent.
3x + y − 3 = 0
3x + y + 2 = 0
Solution
Step 1:
3x + y − 3 = 0
Writing the equation in y = mx + b form,
y = −3x + 3 Equation
3x + y + 2 = 0
y = −3x − 2 Equation
Since the slopes
are the same, the
lines are parallel.
Step 2:
Writing the equation in y = mx + b form,
Equation
y = −3x + 3
:
m = −3 and b = 3
–3
Therefore, the slope is 1 and the y-intercept is (0, 3).
Step 3:
y = −3x − 2
:
Equation
m = − 3 and b = − 2
–3
Therefore, the slope is 1 and the y-intercept is (0, −2).
Graphing equations
using the slope and y-intercept form:
and
Y
5
4
3
(0, 3)
2
3x + y + 2 = 0
1
X
-5
-4
-3
-2
-1
-1
-2
1
2
3
4
5
3x + y – 3 = 0
(0, −2)
-3
-4
-5
The lines have the same slopes but different y-intercepts. Therefore, the lines are parallel and distinct;
i.e., they have no solutions.
The system is inconsistent (no solution) and the equations are independent (lines are not coincident).
9.3 Solving Systems of Linear Equations with Two Variables, Graphically
306
Example 9.3-c
Analyzing System of Equations
Without graphing, determine whether each system has one solution, no solution, or many solutions.
Solution
(i)
4x + y = 9
(ii) 2x + y = 5
2y + x = 4
(iii)
2x + 4y = 16
2x − 3y + 6 = 0
6x − 9y = −18
(i)
Rewrite the equations in slope and y-intercept form (y = mx + b):
4x + y = 9
y = −4x + 9 Slope (m) = −4, y-intercept (b) = 9
2x + y = 5
y = −2x + 5 Slope (m) = −2, y-intercept (b) = 5
The slopes of the lines are different (−4 and −2).
Therefore, the lines are not parallel. They will intersect at one point. The system will have one solution.
(ii)
Rewrite the equations in slope and y-intercept form (y = mx + b):
2y + x = 4
2y = −x + 4
1
y=− x+2
2
1
Slope (m) = − , y-intercept (b) = 2
2
2x + 4y = 16
4y = −2x + 16
1
y=− x+4
2
1
Slope (m) = − , y-intercept (b) = 4
2
1
The slopes of the lines are the same (− ) but the y-intercepts are different.
2
Therefore, the lines are parallel but distinct. The system will have no solutions.
(iii) Rewrite the equations in slope and y-intercept form (y = mx + b):
2x − 3y + 6 = 0
3y = 2x − 6
2
y= x+2
3
Slope (m) =
2
, y-intercept (b) = 2
3
Slope (m) =
2
, y-intercept (b) = 2
3
6x − 9y = −18
9y = 6x + 18
2
y= x+2
3
The slopes of the lines are the same ( 2 ) and the y-intercepts are also the same (2).
3
Therefore, the two lines coincide. The system will have many solutions.
Chapter 9 | Graphs and Systems of Linear Equations
307
9.3 Exercises
Answers to odd-numbered problems are available at the end of the textbook.
For Problems 1 to 8, without graphing, using the slope property of parallel and perpendicular lines, identify whether each of
the pairs of the following lines are parallel, perpendicular, or intersecting.
1.
y=x+1
2.
4x + 4y = –1
6x – 5y = 10
y = – 6 x – 12
5
3x – 2y = –12
x – 3y = – 60
y= 1 x–4
3
4.
5.
2x + 5y = –5
y= 5 x–4
2
6.
7x + 4y = 16
y=– 4 x+3
7
7.
3x + 2y = –24
y= 3 x+3
2
8.
3x – 2y = – 6
y = 3 x – 16
2
3.
2x + 3y = –12
For Problems 9 to 16, without graphing, determine whether each system has one solution, no solution, or many solutions.
9.
3x + 4y = 4
2x + y = 6
11. x – y = 1
2x + y = 5
13. 3y – 2x = 1
12y – 8x = –4
15. 2x = 3y – 1
8x = 12y + 4
10. 3x − 2y = 6
x + 2y = 6
12. x + y = 7
2x – y = 8
14. 2x – y – 4 = 0
6x – 3y + 12 = 0
16. x + 2y = 5
x + 4y = 9
17. Find the value of ‘A’ for which the lines Ax – 2y – 5 = 0 and 8x – 4y + 3 = 0 are parallel.
18. Find the value of ‘B’ for which the lines 3x + 2y + 8 = 0 and 6x – By – 3 = 0 are parallel.
19. Find the value of ‘A’ for which the lines x + 3y + 1 = 0 and Ay + 2x + 2 = 0 are coincident.
20. Find the value of ‘B’ for which the lines y = Bx + 3 and x – 2y + 6 = 0 are coincident.
For Problems 21 to 40, solve the system of equations by graphing, then classify the system of equations as consistent or
inconsistent and the equations as dependent or independent.
21. y = 3x − 2
y = −7x + 8
23. y = −x + 2
2y = −2x + 6
25. y = −4x + 7
2y + 8x = 14
27. 3x − y − 8 = 0
6x − 2y − 1 = 0
29. y − x + 1 = 0
y + 2x − 5 = 0
31. 5x + y + 9 = 0
x − 3y + 5 = 0
22. y = 3x + 9
y=x−4
24. y = 2x + 6
2y = −3x + 6
26. y = −2x + 3
2y + 4x = 6
28. y = 2x − 4
3y = −2x + 4
30. y = x −2
3y = −2x + 9
32. 3x − 2y + 1 = 0
y + 4x − 6 = 0
9.3 Solving Systems of Linear Equations with Two Variables, Graphically
308
33. 3x + 2y = −4
y+ 3x+2=0
2
34. 2x − y = 6
35. 4x − 2y = 6
36. 2y − x − 6 = 0
1
y= x+3
2
6x − 3y = 15
−2y + 4x − 8 = 0
37. x − y = 6
38. 4x − 8y = 0
2x + y = 3
2x − 4y = −8
1
x+1=0
2
4y − x − 4 = 0
40. y = x + 5
39. y +
x + 2y = 10
9.4 Solving Systems of Linear Equations
with Two Variables, Algebraically
Introduction
Solving systems of linear equations using algebraic methods is the most accurate for the following
reasons:
1. It eliminates graphing errors.
2. It provides the exact answer for systems of equations that have fractions or that have fractional
answers.
There are two algebraic (non-graphical) methods for solving systems of linear equations. They are:
■■ Substitution method
■■ Elimination method
Substitution Method
The substitution method is preferable if either one of the equations in the system has a variable with a
coefficient of 1 or −1. The substitution method is easier to use when one variable stands alone on one side
of the equation.
In this method, the following steps are used to solve systems of two linear equations with two variables:
Step 1:
Rewrite both equations in the form of Ax + By = C, where A, B, and C are integers.
Step 2:
Choose the simplest equation from Step 1 and solve to find an expression for one variable
in terms of the other variable.
Step 3:
Substitute the expression for the variable from Step 2 into the other equation (the one not
used in Step 2). This will result in an equation for one of the variables.
Step 4:
Solve the equation in Step 3 for the one variable.
Step 5:
Solve for the other variable using any one of the equations in Step 1 and by substituting
with the value of the known variable.
The answer for the two variables found in Step 4 and Step 5 will be the solution to the given
system of equations.
Step 6:
Check if the solution obtained for the variables is true by substituting these values in each
of the original equations. If the solution satisfies the equations, then it is the solution to the
given systems of equations.
Note: In Step 2, if possible, it is best to select the equation in which the coefficient on any one of the
variables is equal to one, since this will make the calculations in ‘Steps 3 and 4’ easier.
Chapter 9 | Graphs and Systems of Linear Equations