Determine the number of solutions to this system of linear equations (0, 1, ∞)

Question

(
0
,

1
,

∞
)
:

y=7x+8
y
=
7
x
+
8

−7x+y=−8

Answer 1

To determine the number of solutions to the given system of linear equations, we first need to analyze the two equations:

1. \( y = 7x + 8 \) (Equation 1)
2. \( -7x + y = -8 \) (Equation 2)

Let's start by rewriting Equation 2 in the slope-intercept form \( y = mx + b \) for easier comparison. We can rearrange it:

\[ y = 7x - 8 \]

Now, we have:

1. \( y = 7x + 8 \)
2. \( y = 7x - 8 \)

Next, we observe that both equations have the same slope \( m = 7 \). Since the slopes are equal, this implies that the lines are parallel.

To find if the lines are the same (and thus have infinitely many solutions), we can check if their y-intercepts are the same. The y-intercepts of the two equations are:

• For the first equation \( y = 7x + 8 \), the y-intercept is \( 8 \).
• For the second equation \( y = 7x - 8 \), the y-intercept is \( -8 \).

Since the y-intercepts are different, the lines are not the same. Therefore, the system of equations has no solution at all, because parallel lines do not intersect.

In conclusion, the number of solutions to the system of linear equations is:

\[ \text{Number of solutions} = 0 \]