Question

Two lines or equations are described in each part. Decide whether each system has one solution, no solution, or infinitely many solutions. Show your work or explain your reasoning.
(a) Line a has a rate of change of 7.
Line b has a rate of change of -7



(b)2x+3y=5.5
4x+6y+11




(c) Line c and line d are parallel

MUST EXPLAIN WHY YOU GOT THAT ANSWER TY WILL GIVE BRAINIEST MAX POINT

Answer 1

To get the solution of a set of equations means to get a point that satisfies both equations.

Part (1):
The first line has a rate of change of 7, this means that slope of first line is 7
The second line has a rate of change of -7, this means that slope of second line is -7
Since the slope of the first line = - slope of the second line, then these two lines are definitely perpendicular to each other.
Two perpendicular lines will meet only in one point. This means that one point only will satisfy both equations (check the image showing perpendicular lines attached below)
Therefore, only one solution exists in this case

Part (2): 
The first given equation is:
2x + 3y = 5.5
The second given equation is:
4x + 6y = 11
If we simplified the second equation we will get: 2x + 3y = 5.5 which is exactly similar to the first equation.
This means that the two given equations represent the same line. 
Therefore, we have infinite number of solutions

Part (3):
We are given that the two lines are parallel. This means that the two lines are moving the same path side by side. Two parallel lines can never intersect. This means that no point can satisfy both equations (check the image showing parallel lines attached below).
Therefore, we have no solutions for this case.

Answer 2

The equation of two lines are said to have no solution if the two lines are parallel to each other. This is identified when the equations of the two lines have the same slope.

The equation of two lines are said to have infinitely many solutions if the two lines are coincide (i.e. the two lines are on top of each other). This is identified when the equations of the two lines when simplified have the same slope and intercept.

The equation of two lines are said to have one solution if the two lines are neither parallel to each other nor coincide. The two lines intersect at exactly one point. This is identified when the equations of the two lines have different slope.

Part A:

Given that line a has a rate of change of 7and line b has a rate of change of -7, the two lines have different slopes and hence the system has one solution.



Part B:

Given

2x+3y=5.5
4x+6y+11

Written in slope-intercept form we have:

$2x+3y=5.5\Rightarrow3y=-2x+5.5 \\ \\ \Rightarrow y=- \frac{2}{3} x+ \frac{11}{6} \ .\ .\ .\ (1) \\ \\ 4x+6y=11\Rightarrow6y=-4x+11 \\ \\ \Rightarrow y=- \frac{2}{3} x+ \frac{11}{6} \ .\ .\ .\ (2)$

As can be seen the equations have the same slope and the same intercept. Thus, the lines representing the equations coincide and hence the system has have infinitely many solutions.



Part C:

Given that line c and line d are parallel, thus the system of equations represented by line c and line d has no solutions.