Determine the number of solutions to this system of linear equations: y=x+5 y=−x+5 (1 point) Responses infinite, 1, 0, 2
The system of linear equations is:
y = x + 5 -------(1)
y = -x + 5 ------(2)
To determine the number of solutions to this system, we can graph the two equations and see where they intersect.
Converting equation (1) to slope-intercept form by subtracting 5 from both sides, we get:
y - 5 = x
y = x + 5
This equation has a slope of 1 (the coefficient of x) and a y-intercept of 5 (the value when x=0).
Converting equation (2) to slope-intercept form by subtracting 5 from both sides, we get:
y - 5 = -x
y = -x + 5
This equation also has a slope of -1 (the coefficient of x) and a y-intercept of 5 (the value when x=0).
Graphing the two equations:
The graph of y = x + 5 is a straight line passing through (0,5) with a positive slope.
The graph of y = -x + 5 is also a straight line passing through (0,5), but with a negative slope.
Since the two lines are not parallel and have different slopes, they intersect at exactly one point. Therefore, this system of linear equations has one solution.
Hence, the answer is 1.
To determine the number of solutions to the system of linear equations, we can compare the slopes and y-intercepts of the two equations.
The first equation, y = x + 5, has a slope of 1 and a y-intercept of 5.
The second equation, y = -x + 5, also has a slope of -1 and a y-intercept of 5.
Since the slopes are different, the lines corresponding to these equations are not parallel, and they intersect at a single point.
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, you can analyze the slopes and intercepts of the given equations.
Let's consider the given system of equations:
1) y = x + 5
2) y = -x + 5
These equations are in slope-intercept form, y = mx + b, where m is the slope and b is the y-intercept.
Comparing the slopes of the two equations, we see that the slopes are different: 1 for the first equation and -1 for the second equation. This means the lines representing these equations are not parallel.
Now, let's analyze the intercepts. Both equations have a y-intercept of 5, which means both lines intersect with the y-axis at the point (0, 5).
Since the lines are not parallel and intersect at a single point, this system of equations has exactly 1 solution.
Therefore, the correct response is 1.