Download Chapter 6.1--Solving Systems of Linear Equations by Graphing

Chapter 6.1--Solving Systems of Linear Equations by Graphing
Chapter 6.1--Solving Systems of Linear Equations by Graphing
A system of equations is a set of two or more equations that we consider at the same
time. A solution to a system of equations is a set of points that are solutions to each of
the individual equations.
3x − 2y  −1
xy  3
Example
is a system of equations. The pair 1, 2 is a solution to this
system because 31 − 22  1 and 1  2  3 are both true.
x − 2y  4
2x  y  3
Question 4: "Is 2, −1 a solution to the system
?"
Test 2, −1 in both equations.
2 − 2−1  4
2 ∗ 2  −1  3
Yes it is a solution.
Question 6: Now you do this question: "Is −1, −1 a solution to the system
x − 4y  3
3x  y  2
?"
No! Because 3−1  −1 is not equal to 2.
Solving Systems of Linear Equations
There are many ways to solve systems of equations. We are going to learn three different
ways in this chapter alone, and you will learn other ways in future algebra classes. Since
each linear equation that we encounter in this chapter can be graphed as a straight line,
we can use this fact to solve equations. Solutions occur where the lines intersect each
other.
© W Clarke
1
10/20/2004
Chapter 6.1--Solving Systems of Linear Equations by Graphing
xy  6
Example Consider the system
x−y  2
. We can write the first equation as
y  −x  6
y  x−2.
and the second equation as
These lines are plotted on the same axes below.
y
6
4
2
0
0
2
4
6
x
-2
Notice that the two lines intersect at the point 4, 2 . Check to see that this is indeed the
only solution to the system of equations.
Definition: If the two lines intersect at exactly one point (like the previous example) we
say the equations are independent. If the lines coincide (are really the same line) they are
dependent and every point on the line is a solution. It they are parallel we say the system
is inconsistent and has no solution.
2x − y  4
xy  5
Question 32: "Solve by graphing:
." [0-4&-3-5]
Now solve each equation for y.
2x − y  4 , Solution is: 2x − 4
x  y  5 , Solution is: 5 − x
y
4
2
0
0
1
2
3
4
x
-2
The solution is 3, 2 . When there is a unique (only one) solution we call the system
independent.
© W Clarke
2
10/20/2004
Chapter 6.1--Solving Systems of Linear Equations by Graphing
xy  5
3x  3y  6
." [-4-4&-1-7]
Question 42: "Solve by graphing:
y  5−x
The two equations are
x  y  2 or y  2 − x
and
5−x
y
10
7.5
5
2.5
0
-5
-2.5
0
2.5
5
x
-2.5
Notice that the two lines are parallel so they never intersect. Therefore there is no
solution. We call this kind of system inconsistent.
2x  6y  6
Question 44: "Solve by graphing:
y  − 13 x  1
." [-4-4&-1-3]
1
2x  6y  6 , Solution is: 1 − 3 x
− 13 x  1
y
3
2
1
0
-4
-2
0
2
4
x
-1
The two lines coincide. They are really the same line. Therefore any point on the line is a
solution. There are infinitely many solutions. We call a system like this dependent.
© W Clarke
3
10/20/2004