Answer:
See below
Explanation:
6. a. x + y = 2
4x + 4y = 8
If we rewrite both in standard slope-intercept format, we will see the issue:
y = -x + 2
4y = -4x + 8
y = -x + 2
These equations are the same. There are an infinite number of solutions. All points intersect. So it does not have a real solution. [Perhaps it has a real doozy of a solution.]
b. 3x - 2y = 4
3x - 2y =10
Rewrite both:
-2y = -3x + 4
y = (3/2)x - 2
---
-2y = -3x + 10
y = (3/2)x - 5
These equations are different, but they have the same slope, (3/2). They are parallel and thus will never intersect. There is no real solution.
c. 3x - 2y = 4
6x - 4y = 8
Rewrite both:
-2y = -3x + 4
y = (3/2)x -2
--
-4y = -6x + 8
y = (3/2)x + 8
Same slope of (3/2). Same answer for parallel lines.
d. 5x + 2y = 2
15x + 6y = 6
Rewrite both:
2y = -15x + 6
y = -(15/2)x + 2
---
6y = -15x + 6
y = -(15/6)x + 1
y = -(5/2)x + 1
These lines have different slopes, so they will intersect. This system has a real solution.
=
7. See attached graph for 7.
It's a surprise to find only one line. But rewrite the equations in standard format to reveal why there is only one line:
3x + 5y = 5
5y = -3x + 5
y = -(3/5)x + 1
---
6x + 10y = 10
10y = -6x + 10
y = -(6/10)x + 1
y = -(3/5)x + 1
These equations are the same: Two coinciding lines, to put it mildly.
8. I interpret "Two lines in a system of linear equation in two variable coincide" as both equations are the same. If that is a correct interpretation, Infinite solutions.
9. All three can be used.
10.
a. 5x + 2y = 2
2x - 5y = 2
One solution
They have different slopes. The first is -(5/2) and the second is (2/5). They are actually perpendicular to each other.
b. 5x + 2y = 2
10x + 4y = 4
Rewrite the 2nd equation by dividing by 2: 5x + 2y = 2
These equations are identical. They have an infinite number of solutions (more than 1)
c. 5x + 2y = 4
15x + 6y = 12
Rewrite the second equation:
d. 5x + 2y = 8
5x + 2y = 6
These equations are identical. They have an infinite number of solutions (more than 1)