Graphing Systems of Equations :
Two equations such as y = x - 3 and y = 2x + 5, together are called systems of equations.
A solution of a system of equations is an ordered pair of numbers that satisfies both equations. A system of two linear equations can have 0, 1, or an infinite number of solutions.
Graph of system |
Number of solutions |
Terminology |
Intersecting lines |
1 |
Consistent and independent |
Coincide lines |
Infinitely many solutions |
Consistent and dependent |
Parallel lines |
0 |
Inconsistent |
To graph the given equations, we need to solve each equations for y in terms of x.
Now we need to assign some random values of x, so we get the values of y. Write them as ordered pairs.
We can plot it into the graph sheet and draw a line by joining the plotted points.
Example 1 :
Graph the following system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.
x + y = 8
x - y = 2
Solution :
y = 8 - x
x |
-1 |
0 |
1 |
y |
9 |
8 |
7 |
Set of ordered pairs :
(-1, 9) (0, 8) (1, 7)
y = x - 2
x |
-1 |
0 |
1 |
y |
-3 |
-2 |
-1 |
Set of ordered pairs :
(-1, -3) (0, -2) (1, -1)
Since the above lines are intersecting in one point, it is consistent and it has only one solution.
Hence the solution is (5, 3).
Example 2 :
Graph the following system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.
2x + 4y = 2
3x + 6y = 3
Solution :
y = (2 - 2x)/4
y = (1-x)/2 -----(1)
y = (3 - 3x)/6
y = (1 - x)/2 -----(2)
Since both equations represent the same line, they are coincident.
Hence they have infinitely many solution.
Example 3 :
Graph the following system of equations. Then determine whether the system has no solution, one solution, or infinitely many solutions. If the system has one solution, name it.
x + y = 4
x + y = 1
Solution :
y = 4 - x -----(1)
y = 1 - x -----(2)
Table of 1^{st} equation
x |
-1 |
0 |
1 |
y |
5 |
4 |
3 |
Set of ordered pairs :
(-1, 5) (0, 4) (1, 3)
y = 1 - x
Table of 2^{nd} equation
x |
-1 |
0 |
1 |
y |
2 |
1 |
0 |
Set of ordered pairs :
(-1, 2) (0, 1) (1, 0)
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