Determine the number of solutions to this system of linear equations:
y=x+5
y=−x+5 (1 point)
Responses
1
1
infinite
infinite
0
0
2
The correct answer is 1 solution.
The two equations given are:
1) y = x + 5
2) y = -x + 5
From equation 1, we can rewrite it as x = y - 5.
Substituting this into equation 2 gives us:
y = -(y - 5) + 5
Simplifying this equation, we get:
y = -y + 10
Combining like terms, we have:
2y = 10
Simplifying further, we get:
y = 5
Substituting this value of y into equation 1 or 2 gives us:
5 = x + 5
x = 0
So the solution to the system of linear equations is x = 0, y = 5.
Since there is only one solution, the answer is 1.
The system of linear equations can be rewritten as follows:
y = x + 5 ...(Equation 1)
y = -x + 5 ...(Equation 2)
To determine the number of solutions, we can graph these equations on a coordinate plane.
The equation y = x + 5 represents a line with a positive slope. The equation y = -x + 5 represents a line with a negative slope.
These lines intersect at the point (0, 5). This intersection point is the solution to the system of linear equations.
Therefore, the number of solutions to the system of linear equations is 1.
To determine the number of solutions to a system of linear equations, we can follow a few steps:
Step 1: Write down the system of linear equations:
y = x + 5
y = -x + 5
Step 2: Since both equations are already in the form y = mx + b, where m is the slope and b is the y-intercept, we can see that the slopes of the two lines are opposite, indicating that they are not parallel.
Step 3: Compare the slopes of the two lines. In this case, the slopes are 1 and -1. Since the slopes are not equal, the lines are not parallel and they will intersect at a single point.
Step 4: Therefore, the number of solutions to this system of linear equations is 1.
So, the correct answer is 1.