Question

Without graphing, answer the questions for the system of equations below.

-3x+y = 2
x+6y = 36
a. Are the graphs of the given equations identical lines, parallel lines, or lines intersecting at a single point?
Identical lines
Parallel lines
Intersecting lines
b. How many solutions does the system have?
No solutions
One solution
Infinite solutions

Answer 1

a. The graphs of the given equations are **parallel lines.**

b. The system of equations has **no solutions** because the parallel lines do not intersect.

Explanation:

a. The two equations can be rewritten in slope-intercept form:

For the first equation:

-3x + y = 2

y = 3x + 2

For the second equation:

x + 6y = 36

6y = -x + 36

y = (-1/6)x + 6

In slope-intercept form, the first equation has a slope of 3, and the second equation has a slope of -1/6. When the slopes of two lines are different (in this case, 3 and -1/6), but they are not vertical or horizontal lines, the lines are parallel. Parallel lines do not intersect because they have different slopes and run in the same direction or are in the same plane but never meet.

b. Since the lines are parallel and do not intersect, there is no point where they share the same coordinates. In a system of linear equations, this implies that there are **no solutions** to the system, as the equations represent lines that do not have a common point.