A consistent system of linear equations has one or more solutions.
A consistent system of linear equations refers to a set of linear equations that can be solved simultaneously, such that there is at least one solution
A consistent system of linear equations refers to a set of linear equations that can be solved simultaneously, such that there is at least one solution. In other words, if the system of equations has a common solution or a set of solutions that satisfy all the equations in the system, it is considered consistent.
To understand this concept better, let’s consider an example. Suppose we have the following system of linear equations:
Equation 1: 2x + 3y = 7
Equation 2: 4x – y = 2
To determine if this system is consistent, we can try to solve it.
One way to solve this system is by using the method of substitution. We can solve Equation 2 for y:
y = 4x – 2
Now, we substitute this value of y into Equation 1:
2x + 3(4x – 2) = 7
Simplifying this equation, we get:
2x + 12x – 6 = 7
14x = 13
x = 13/14
Plugging this value of x back into Equation 2, we can find y:
y = 4(13/14) – 2
y = 26/7 – 2
y = -4/7
So, we have found a unique solution for this system of equations: x = 13/14 and y = -4/7.
Since there exists a solution, this system of linear equations is consistent.
In summary, a consistent system of linear equations has one or more solutions, indicating that there is a set of values that satisfy all the equations in the system.
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