Determine the number of solutions to this system of linear equations: y=x+5 y=−x+5 (1 point)
There is a unique solution to this system of linear equations because the two equations represent two lines that intersect at a single point.
To determine the number of solutions to this system of linear equations, we can graph the two equations and see where they intersect.
First, let's rearrange the equations to the slope-intercept form (y = mx + b) where m represents the slope and b represents the y-intercept.
For the first equation, y = x + 5, the slope is 1, and the y-intercept is 5.
For the second equation, y = -x + 5, the slope is -1, and the y-intercept is 5.
To graph these equations, we can plot the y-intercepts and use the slopes to determine additional points.
For the first equation, y = x + 5, starting at the y-intercept (0,5), we can move one unit up and one unit to the right to get the point (1,6). We can repeat this process to get additional points to graph the line.
For the second equation, y = -x + 5, starting at the y-intercept (0,5), we can move one unit up and one unit to the left to get the point (-1,6). We can repeat this process to get additional points to graph the line.
Plotting the points and drawing the lines, we get:
```
y
|
|
|
| x + 5
|
|
| -x + 5
| _ _ _ _ _ _ _ _ _
-5 -4 -3 -2 -1 0 1 2 3
x
```
From the graph, we can see that the two lines intersect at the point (0, 5). This means that the system of equations has one solution.
Therefore, the number of solutions to this system of linear equations is 1.
To determine the number of solutions to this system of linear equations, we can compare the slopes of the two lines. The slopes of these lines are m1 = 1 and m2 = -1, which means that the lines are not parallel.
Since the slopes are different, the two lines will intersect at a unique point. Therefore, the system of equations has exactly one solution.
To find the solution, we can solve the system of equations by setting the two equations equal to each other:
x + 5 = -x + 5.
Rearranging the equation, we get:
2x = 0.
Simplifying further, we find:
x = 0.
Now, substituting this value of x back into either of the original equations, we can find the corresponding value of y:
y = 0 + 5 = 5.
Hence, the solution to the system of linear equations is x = 0 and y = 5. The system has one unique solution.