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

Question

y=x+5
y
=
x
+
5

y=−x+5
y
=
−
x
+
5
(1 point)
Responses

infinite
infinite

0
0

1
1

2

Answer 1

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

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

We can analyze the equations:

1. The first equation \( y = x + 5 \) represents a line with a slope of 1, which means it rises as \( x \) increases.
2. The second equation \( y = -x + 5 \) represents a line with a slope of -1, which means it falls as \( x \) increases.

To find the point of intersection, we can set the two equations equal to each other:

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

Now, solve for \( x \):

1. Add \( x \) to both sides: \[ 2x + 5 = 5 \]

2. Subtract 5 from both sides: \[ 2x = 0 \]

3. Divide by 2: \[ x = 0 \]

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

Using the first equation: \[ y = 0 + 5 = 5 \]

Thus, the point of intersection is \( (0, 5) \), which means the two lines intersect at a single point.

Therefore, the number of solutions to the system of equations is:

1