A system of equations is said to have no solution when there is no set of values for the variables that satisfies all of the equations in the system simultaneously. Graphically, this means that the lines representing the equations do not intersect; they are parallel. Because they do not intersect, there is no point (x, y) that satisfies both equations at the same time.
Graphical Explanation
To visualize this, consider two linear equations in a two-dimensional Cartesian coordinate system. If you were to graph each equation:
- If the lines are parallel, they have the same slope but different y-intercepts.
- Because they never meet, there are no points (x, y) that lie on both lines, and thus, no solution exists.
Example with Plugging in Values
Let's assume we have a system of linear equations, for example:
- \( y = 2x + 1 \)
- \( y = 2x - 3 \)
Now, let's check the points \((1, 1)\) and \((0, 2)\):
-
For the first equation:
- Plugging in \( x = 1 \): \[ y = 2(1) + 1 = 3. \]
- So, the point \((1, 1)\) does not satisfy the first equation because \( y \neq 3 \).
-
For the second equation:
- Plugging in \( x = 1 \): \[ y = 2(1) - 3 = -1. \]
- The point \((1, 1)\) does not satisfy the second equation either because \( y \neq -1 \).
Now, checking the point \((0, 2)\):
-
For the first equation:
- Plugging in \( x = 0 \): \[ y = 2(0) + 1 = 1. \]
- The point \((0, 2)\) does not satisfy the first equation because \( y \neq 1 \).
-
For the second equation:
- Plugging in \( x = 0 \): \[ y = 2(0) - 3 = -3. \]
- The point \((0, 2)\) does not satisfy the second equation either because \( y \neq -3 \).
Conclusion
Since neither point \((1, 1)\) nor \((0, 2)\) satisfies both equations, it confirms that there are no intersection points (solutions) for the two lines, thus showing that the system has no solution. Graphically, you would observe two parallel lines on a graph, verifying that they do not intersect.
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