Determine the number of solutions to this system of linear equations: y=x+5 y=−x+5
The two equations are y = x + 5 and y = -x + 5.
To determine the number of solutions to this system of linear equations, we can graph the two lines represented by these equations.
The first equation, y = x + 5, has a slope of 1 and a y-intercept of 5. So, one point on this line is (0, 5).
The second equation, y = -x + 5, has a slope of -1 and a y-intercept of 5. So, one point on this line is also (0, 5).
When we graph these lines, we see that they intersect at the point (0, 5).
Therefore, there is only one solution to this system of linear equations.
To determine the number of solutions to this system of linear equations, we can graph the equations and see where they intersect.
The first equation, y = x + 5, has a slope of 1 and a y-intercept of 5. This means that if we plot this equation on a graph, it will have a positive slope and will intersect the y-axis at the point (0, 5).
The second equation, y = -x + 5, also has a slope of -1 and a y-intercept of 5. This means that if we plot this equation on a graph, it will have a negative slope and will also intersect the y-axis at the point (0, 5).
Since both equations have the same y-intercept and intersect at the same point, this means that they are coincident lines. In other words, they are the same line.
Therefore, the system of linear equations has infinitely many solutions, as every point on the line y = x + 5 and y = -x + 5 is a solution to the system.
To determine the number of solutions to a system of linear equations, we can compare the slopes and intercepts of the equations.
Let's examine the given system of equations:
y = x + 5 ------- Equation 1
y = -x + 5 ------- Equation 2
In Equation 1, the coefficient of x is 1, indicating that the slope is 1. In Equation 2, the coefficient of x is -1, indicating that the slope is -1. Since the slopes are different, the two lines are not parallel and should intersect at some point.
Now, let's compare the intercepts. In Equation 1, the y-intercept is 5, and in Equation 2, the y-intercept is also 5. Since the y-intercepts are the same, the lines are parallel.
Therefore, the system of equations represents two parallel lines that do not intersect. Since they do not intersect, there are no common solutions to the system of equations. In other words, the system is inconsistent, and the number of solutions is 0.