Infinitely many solutions:
y = –2x – 4
y + 4 = –2x |
The solution of a system is the point of intersection, if it exists.
One easy way to find the solution of a system of equations is simply to graph the two lines, and then read the coordinates of the point of intersection.
For example, in the second graph shown above, the intersection point is (–2, 1).
One drawback of this method is that you can't be sure you have the coordinates exactly. You can't be really sure that the intersection point is (–2, 1) and not (–2.001, 0.999). To be sure, you should check the point algebraically in both equations.
Substituting x = –2, y = 1 in y = 0.5x + 2, we get:
1 = 0.5(–2) + 2
1 = –1 + 2
1 = 1
Substituting x = –2, y = 1 in y = –2x – 3, we get:
1 = –2(–2) – 3
1 = 4 – 3
1 = 1
So, in this case, the solution checks out.
Is the ordered pair (1, 3) a solution of the given system of equations?
3x + 5y = 18
x – 3y = –8
Is the ordered pair (5, 4) a solution of the given system of equations?
5t – 4s = 0
t + 3s = 15
Is the ordered pair (–2, 2) a solution of the given system of equations?
x = –2
x – y = –4
Is the ordered pair (15, 5) a solution of the given system of equations?
2x – y = 25
Determine whether the ordered pair (4, 6) is a solution of the given system of linear equations.
–16x + 5y = 5
2x – y = 5
Estimate the solution of the given linear system using the graph and then verify the solution algebraically.
–x + 3y = 7
x + 5y = 17
Estimate the solution of the given linear system using the graph and then verify the solution algebraically.
x + y = 2
–4x + y = –8
Use the graph shown to find whether the following system of equations has one solution, no solution, or infinitely many solutions. If the system has one solution, name it.
y = x – 7
Use the graph drawn to find whether each system of equations has one solution, no solution, or infinitely many solutions. If the system has one solution, name it.
Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely any solutions. If the system has a solution, name it.
y = –2x
y = x – 9
Graph the system of equations. Determine whether the system has one solution, no solution, or infinitely any solutions. If the system has a solution, name it.
y = x
y = 8 – x
Graph the lines to solve the system.
y = –x – 2
y
= 3x + 2
Graph the lines to solve the system.
x – y = 4
3x + y = 0
Graph the lines to solve the system.
4x – 8y = 0
–x
+ 2y = –6
Graph the given linear system and estimate their solution. Then verify the solution algebraically.
y = –7
x = 7
Graph the given linear system and estimate their solution. Then verify the solution algebraically.
3x – 4y = 8
x = –4
Graph the given linear system and estimate their solution. Then verify the solution algebraically.
x
– y = 1
6x – 5y = 0
Graph the given linear system and estimate their solution. Then verify the solution algebraically.
y
= –3x + 9
y = 3x + 3
The people of the island nation of Iceland were converted to Christianity from the old Norse polytheistic religion starting around 1000 A.D. Suppose  represents the percent of Icelanders professing Christianity at time t, and  represents the percent that was non–Christian. Find the time at which there were equal numbers of Christians and non–Christians in Iceland. (Hint: let t = 0 represent the year 1000.)
Find the area of the triangle bounded by the following lines:
y–axis
3x
+ 4y = 20
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