Respuesta :
System of linear equation have no solutions both when both the lines are parallel(not intersecting or coincident).
Percy is wrong since system will have no solution for any real numbers except k = 7, because at k = 7, both lines become same, thus having infinite solutions.
When does a system of linear equation have no solution?
When both the lines(represented by both the equations) are parallel and not coincident, then that system of linear equations have no solutions as there is no common point on both line.
If they are coincident(lying over each other), then there will have infinite solution since in that case, they will have infinite points in common.
Thus, if two lines are not in any of above case, they intersect at single point and in that case, the considered system of equation has unique single solution.
When will the given system of equation have no solution?
The first equation is
this can be rewritten as:
[tex]6x + 4y = 14\\ \\ \text{Dividing by 2 on both sides}\\ \\ 3x + 2y = 7[/tex]
Writing in slope-intercept form:
[tex]3x + 2y = 7\\
2y = 7-3x\\\\
y = -\dfrac{3}{2}x + \dfrac{7}{2}[/tex]
The second equation is
[tex]3x + 2y =k[/tex]
If k = 7, then both equation become same thus infinite solution.
[tex]3x + 2y = k\\ 2y = k - 3x\\\\ y = \dfrac{k}{2} -\dfrac{3}{2}x\\\\ y = -\dfrac{3}{2}x +\dfrac{k}{2}[/tex]
If k is not 7, then the y intercept of both the lines will be different but slope will be same. The same slope denotes lines are parallel and different y-intercept shows that the lines are not same(not coincident).
Thus, there are infinite solutions for all values of k except 7.
Thus, Percy is wrong since system will have no solution for any real numbers except k = 7, because at k = 7, both lines become same, thus having infinite solutions.
Learn more about solutions of system of equations here:
https://brainly.com/question/16405744
Answer:
Sample Response: k = 7 proves the equations are equivalent and, therefore, have infinitely many solutions. That is illustrated by the fact that multiplying 2 by 3x +2y = 7 will create the equivalent of 6x + 4y =14.
Step-by-step explanation:
