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When does a system of linear equations in two variables have a unique or one solution?
simultaneous equations
What does a air of linear equations having a unique solution represent graphically?
Presumably the question concerned a PAIR of linear equations! The answer is two straight lines intersecting at the point whose coordinates are the unique solution.
Does every pair of linear simultaneous equations have a unique solution?
So, take the case of two parallel lines, there is no solution at all. Now look at two equations that represent the same line, they have an infinite number of solutions. The solution is unique if and only if there is a single point of intersection. That point is the solution.
Is it possible for a system of three linear equations to have one solution?
Yes, it is possible for a system of three linear equations to have one solution. This occurs when the three equations represent three planes that intersect at a single point in three-dimensional space. For this to happen, the equations must be independent, meaning no two equations are parallel, and not all three planes are coplanar. If these conditions are met, the system will yield a unique solution.
What is the difference between the solution of a system of linear inequalities and the solution of a system of linear equations?
The solution of a system of linear equations consists of specific points where the equations intersect, typically yielding a unique point, infinitely many points, or no solution at all. In contrast, the solution of a system of linear inequalities represents a region in space, encompassing all points that satisfy the inequalities, often forming a polygonal shape in two dimensions. While equations define boundaries, inequalities define areas that can include multiple solutions. Thus, the nature of their solutions differs fundamentally: precise points versus expansive regions.
When does a system of linear equations in two variables have a unique or one solution?
simultaneous equations
What does a air of linear equations having a unique solution represent graphically?
Presumably the question concerned a PAIR of linear equations! The answer is two straight lines intersecting at the point whose coordinates are the unique solution.
What is a unique solution in linear equations?
This is the case when there is only one set of values for each of the variables that satisfies the system of linear equations. It requires the matrix of coefficients. A to be invertible. If the system of equations is y = Ax then the unique solution is x = A-1y.
What is cramer rule?
In linear algebra, Cramer's rule is an explicit formula for the solution of a system of linear equations with as many equations as unknowns, valid whenever the system has a unique solution.
What are the applications of cramer's rule?
Cramer's rule is applied to obtain the solution when a system of n linear equations in n variables has a unique solution.
What is the condition for unique solution of linear equation?
The equations are consistent and dependent with infinite solution if and only if a1 / a2 = b1 / b2 = c1 / c2.
Does every pair of linear simultaneous equations have a unique solution?
So, take the case of two parallel lines, there is no solution at all. Now look at two equations that represent the same line, they have an infinite number of solutions. The solution is unique if and only if there is a single point of intersection. That point is the solution.
Is it possible for a system of three linear equations to have one solution?
Yes, it is possible for a system of three linear equations to have one solution. This occurs when the three equations represent three planes that intersect at a single point in three-dimensional space. For this to happen, the equations must be independent, meaning no two equations are parallel, and not all three planes are coplanar. If these conditions are met, the system will yield a unique solution.
It is impossible for a system of two linear equations to have exactly one solution. True or False?
False, think of each linear equation as the graph of the line. Then the unique solution (one solution) would be the intersection of the two lines.
What is the difference between the solution of a system of linear inequalities and the solution of a system of linear equations?
The solution of a system of linear equations consists of specific points where the equations intersect, typically yielding a unique point, infinitely many points, or no solution at all. In contrast, the solution of a system of linear inequalities represents a region in space, encompassing all points that satisfy the inequalities, often forming a polygonal shape in two dimensions. While equations define boundaries, inequalities define areas that can include multiple solutions. Thus, the nature of their solutions differs fundamentally: precise points versus expansive regions.
How many solutions did this linear system have?
To determine how many solutions a linear system has, we need to analyze the equations involved. A linear system can have one unique solution, infinitely many solutions, or no solution at all. This is usually assessed by examining the coefficients and constants of the equations, as well as using methods like substitution, elimination, or matrix analysis. If the equations are consistent and independent, there is one solution; if they are consistent and dependent, there are infinitely many solutions; and if they are inconsistent, there are no solutions.
When you graph a system of linear equations why does the intersection of the two lines represent the solution of the system?
The intersection of two lines in a graph of a system of linear equations represents the solution because it is the point where both equations are satisfied simultaneously. At this point, the x and y coordinates meet the conditions set by both equations, meaning that the values of x and y make both equations true. Hence, the intersection point is the unique solution to the system, assuming the lines are not parallel or coincident.
