Pair Of Linear Equations In Two Variables

Explanation
A linear equation in two variables represents a straight line graphically. This is because a linear equation has a degree of 1, meaning that the highest power of the variables is 1. A straight line can be represented by an equation in the form y = mx + b, where m is the slope and b is the y-intercept. Therefore, the correct answer is a straight line.

Explanation
If a pair of linear equations has a unique solution, it means that the lines represented by these equations intersect at a single point. This is because the unique solution represents the coordinates of the point where the lines intersect. Therefore, the correct answer is "intersecting".

Explanation
If a pair of linear equations in two variables have no solution, it means that the lines represented by these equations do not intersect. In other words, they are parallel to each other. Therefore, the correct answer is "parallel."

Explanation
For the given system of equations to have no solution, the slopes of the two lines formed by the equations must be equal. In this case, the slope of the first equation is -2/6 = -1/3, and the slope of the second equation is -4/k. Therefore, the slope of the second equation must be equal to -1/3. Setting -4/k = -1/3 and solving for k, we get k = 12. So, for k = 12, the system of equations 2x+6y=5 and 4x+12y=7 will have no solution.

Explanation
The given system of equations is x+y=4 and x-y=6. To solve this system, we can use the method of elimination. By adding the two equations, we get 2x=10, which gives us x=5. Substituting this value of x into either of the original equations, we find y=-1. Therefore, the solution to the system of equations is x=5 and y=-1.

Explanation
A pair of lines may have an infinite number of solutions when they are coincident or overlapping. This means that the two lines lie on top of each other and intersect at every point along the line. In this case, any point on the line is a solution to the system of equations represented by the lines. Therefore, the statement "A pair of lines may have an infinite number of solutions" is true.

Explanation
The given equations represent coincident lines when their slopes are equal and their y-intercepts are also equal. Comparing the equations, we can see that the slope of the first equation is 3 and the slope of the second equation is 6. To make the slopes equal, we need to set the coefficient of y in the second equation equal to 3. Therefore, k = 3/2. However, none of the given options match this value. Therefore, the correct answer is not available.

Explanation
The given pair of linear equations is inconsistent, meaning that there are no values of x and y that satisfy both equations simultaneously. This can be determined by observing that the second equation is a multiple of the first equation, which means that the lines represented by these equations are parallel and will never intersect. Therefore, there are no solutions to this pair of equations, resulting in 0 solutions.

Explanation
The lines in the graph represent a unique solution. This means that there is only one point of intersection between the lines, indicating that there is only one solution to the system of equations represented by the graph.

Explanation
For the equation 2x+y+c to pass through the origin (0,0), we substitute x=0 and y=0 into the equation. This gives us 2(0)+0+c=0+c=c=0. Therefore, the value of c for which the equation passes through the origin is c=0.