Determine the number of solutions to this system of linear equations:

Question

y=x+5

y=−x+5
(1 point)
Responses

infinite
infinite

2
2

0
0

1

Answer 1

To determine the number of solutions to the system of linear equations given by:

1. \( y = x + 5 \)
2. \( y = -x + 5 \)

we can analyze the equations graphically or algebraically.

1. Graphical Approach:
   • The first equation, \( y = x + 5 \), represents a straight line with a slope of 1 that intersects the y-axis at (0, 5).
   • The second equation, \( y = -x + 5 \), represents a straight line with a slope of -1 that also intersects the y-axis at (0, 5).

Both lines intersect at the same point, which is evident from setting the two equations equal to each other to find the intersection:

\[ x + 5 = -x + 5 \]

Solving for \( x \):

\[ x + x = 5 - 5 \] \[ 2x = 0 \] \[ x = 0 \]

Now, substituting \( x = 0 \) into one of the original equations to find \( y \):

\[ y = 0 + 5 = 5 \]

Thus, both lines intersect at the single point (0, 5).

2. Algebraic Approach: As we already determined, both lines intersect at one unique point.

Given this analysis, the system of equations has exactly 1 solution: the point (0, 5).

Thus, the answer is:

1.