How many solutions does a consistent and dependent system of linear equations have?
Are you sure it said "dependent"? Dependent means the lines are the same. They intersect everywhere like for example
x + y = 2
2x + 2y = 4
The lines cross at an infinite number of points and there are an infinite number of solutions.`
Do you mean dependent of independent equations?
It depends upon how many equations and variables there are. If any of the equations are dependent on (derivable from) others, and there are at least as many variables as equations, then the system of equations in underdetermined and there are an infinite number of solutions
nope that's what the question is on the paper....
A consistent and dependent system of linear equations has infinitely many solutions. Let's break down how we can arrive at this conclusion:
A system of linear equations is consistent if it has at least one solution, meaning there is a set of values that satisfies all the equations simultaneously. On the other hand, a system is dependent if one or more of its equations can be expressed as a linear combination of the others.
When a system is both consistent and dependent, it means that one or more equations in the system can be derived from combinations of the other equations. Therefore, the equations in the system are not linearly independent.
To determine how many solutions this type of system has, we can consider an example where we have two equations:
Equation 1: 2x + 3y = 8
Equation 2: 4x + 6y = 16
By observing Equation 1, we can see that if we multiply it by 2, we get Equation 2. In other words, Equation 2 is simply a scaled version of Equation 1. Hence, these equations are linearly dependent.
If we graph the equations on a coordinate plane, we will notice that the two lines representing the equations will be coinciding or overlapping. This implies that they intersect at infinitely many points, which translates to infinitely many solutions.
Therefore, a consistent and dependent system of linear equations has infinitely many solutions.