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Can a system of two linear equations have exactly two solutions explain?
No, a system of two linear equations cannot have exactly two solutions. In a two-dimensional space, two linear equations can either intersect at one point (one solution), be parallel (no solutions), or be the same line (infinitely many solutions). Therefore, it is impossible for a system of two linear equations to have exactly two solutions.
Why could a system of linear equations have two solutions?
A system of linear equations cannot have two distinct solutions if it is consistent and defined in a Euclidean space. If two linear equations intersect at a single point, they have one solution; if they are parallel, they have no solutions. However, if the equations are dependent, meaning one equation is a multiple of the other, they represent the same line and thus have infinitely many solutions, not just two. Therefore, in standard scenarios, a system of linear equations can either have one solution, no solutions, or infinitely many solutions, but not exactly two.
If a system of linear of linear equations has infinitely many solutions what does this mean about the two lines?
If a system of linear equations has infinitely many solutions, it means that the two lines represented by the equations are coincident, meaning they lie on top of each other. This occurs when both equations represent the same line, indicating they have the same slope and y-intercept. As a result, any point on the line is a solution to the system.
What does it mean if there are an infinite number of solutions for a system of linear equations?
It means that the equations are actually both the same one. When they're graphed, they both turn out to be the same line.
It is possible for a system of linear equations to have an infinite number of solutions.?
Yes, a system of linear equations can have an infinite number of solutions when the equations represent the same line or when they are dependent on each other. This typically occurs in systems with fewer independent equations than variables, leading to free variables that allow for multiple solutions. In such cases, the solutions can be expressed in terms of parameters, indicating a whole line or plane of solutions rather than a single point.
What are two or more linear equations using the same variables called?
A system of linear equations.
What are equations with the same solution?
Equations with the same solution are called dependent equations, which are equations that represent the same line; therefore every point on the line of a dependent equation represents a solution. Since there is an infinite number of points on a line, there is an infinite number of simultaneous solutions. For example, 2x + y = 8 4x + 2y = 16 These equations are dependent. Since they represent the same line, all points that satisfy either of the equations are solutions of the system. A system of linear equations is consistent if there is only one solution for the system. A system of linear equations is inconsistent if it does not have any solutions.
Why Two dependent simultaneous linear equations have infinite solutions?
Two dependent linear equations are effectively the same equation - with their coefficients scaled up or down.
What does a graph with infinitely many solutions look like?
they have same slop.then two linear equations have infinite solutions
What is equations that have the same solutions?
Simultaneous equations have the same solutions.
If a system of linear of linear equations has infinitely many solutions what does this mean about the two lines?
If a system of linear equations has infinitely many solutions, it means that the two lines represented by the equations are coincident, meaning they lie on top of each other. This occurs when both equations represent the same line, indicating they have the same slope and y-intercept. As a result, any point on the line is a solution to the system.
What are the three types of possible solutions to a system of equations?
If the equations are linear, they may have no common solutions, one common solutions, or infinitely many solutions. Graphically, in the simplest case you have two straight lines; these can be parallel, intersect in a same point, or actually be the same line. If the equations are non-linear, they may have any amount of solutions. For example, two different intersecting ellipses may intersect in up to four points.
What does it mean if there are an infinite number of solutions for a system of linear equations?
It means that the equations are actually both the same one. When they're graphed, they both turn out to be the same line.
It is possible for a system of linear equations to have an infinite number of solutions.?
Yes, a system of linear equations can have an infinite number of solutions when the equations represent the same line or when they are dependent on each other. This typically occurs in systems with fewer independent equations than variables, leading to free variables that allow for multiple solutions. In such cases, the solutions can be expressed in terms of parameters, indicating a whole line or plane of solutions rather than a single point.
What is the definition of Simultaneous Linear Equations?
A system of linear equations is two or more simultaneous linear equations. In mathematics, a system of linear equations (or linear system) is a collection of linear equations involving the same set of variables.
Does every pair of linear simultaneous equations have a solution?
Actually not. Two linear equations have either one solution, no solution, or many solutions, all depends on the slope of the equations and their intercepts. If the two lines have different slopes, then there will be only one solution. If they have the same slope and the same intercept, then these two lines are dependent and there will be many solutions (infinite solutions). When the lines have the same slope but they have different intercept, then there will be no point of intersection and hence, they do not have a solution.
What are equations that have the same answer?
Simultaneous equations have the same solutions
