Systems of Linear Equations
A system of equations consists of two or more equations, each of which
contains at least one variable. Here are three examples:
| System 1 |
System 2 |
System 3 |
|
3x + y
4x - 2y |
= -5 = 7 |
-7x + 9y
3y |
= 0 = 8 |
5x - 4y
y |
= 11 = 3x + 1 |
Each system is called a linear system in two variables. This is because
the graph of each equation is linear (that is, the graph is a straight line)
and two variables are involved.
An ordered pair, (x, y), is a solution of a linear system of equations in
two variables if the ordered pair makes each equation true. An ordered
pair that is a solution is said to satisfy the system.
Example 1
Determine if (5, -2) is a solution of this system.
| 3x - 4y x + y |
= 23 = 3 |
First equation Second equation |
Solution
In each equation, replace x with 5 and y with -2. Then simplify.
| |
First equation |
|
Second equation |
| Is
Is
Is |
3x
3(5)
15 |
- -
+ |
4y
4(-2)
8
23 |
= 23 = 23 ?
= 23 ?
= 23 ? Yes |
Is
Is |
x
(5) |
+ + |
y
(-2)
3 |
= 3 = 3 ?
= 3 ? Yes |
Since (5, -2) satisfies each equation, it is a solution of the system.
The solution can be written as x = 5 and y = -2, or simply (5, -2).
|
