Let's solve the given system of equations step by step and determine if there is a solution or not.
The two given equations are:
1. [tex]\( y = -x + 8 \)[/tex]
2. [tex]\( y = -x + 10 \)[/tex]
To analyze these equations, let's start by graphing them.
1. Graph of [tex]\( y = -x + 8 \)[/tex]:
- This is a straight-line equation in slope-intercept form ( [tex]\( y = mx + b \)[/tex]), where [tex]\( m \)[/tex] is the slope and [tex]\( b \)[/tex] is the y-intercept.
- Slope [tex]\( m = -1 \)[/tex]
- Y-intercept [tex]\( b = 8 \)[/tex]
To graph this line, plot the y-intercept (0, 8) and use the slope to find another point. Since the slope is -1, for each step of 1 unit right (along x-axis), we move 1 unit down (along y-axis).
Another point could be:
- Start at (0, 8), move 1 unit right: [tex]\( x = 1 \)[/tex]
- Move 1 unit down: [tex]\( y = 7 \)[/tex]
- Point: [tex]\( (1, 7) \)[/tex]
With these points [tex]\((0,8)\)[/tex] and [tex]\((1,7)\)[/tex], draw the line extending through them.
2. Graph of [tex]\( y = -x + 10 \)[/tex]:
- This is another straight-line equation in the same form, with different parameters.
- Slope [tex]\( m = -1 \)[/tex]
- Y-intercept [tex]\( b = 10 \)[/tex]
To graph this line, plot the y-intercept (0, 10) and use the slope to determine another point. Similarly:
- Start at (0, 10), move 1 unit right: [tex]\( x = 1 \)[/tex]
- Move 1 unit down: [tex]\( y = 9 \)[/tex]
- Point: [tex]\( (1, 9) \)[/tex]
With these points [tex]\((0,10)\)[/tex] and [tex]\((1,9)\)[/tex], draw the line extending through them.
Analysis of the Intersection:
Upon examining the two equations:
- Both lines have an identical slope of -1, indicating that they are parallel.
- The y-intercept for the first line is 8, and for the second line is 10.
Since parallel lines never intersect (they have the same slope but different y-intercepts), there is no point [tex]\((x, y)\)[/tex] that satisfies both equations simultaneously.
Thus, the conclusion for the system of equations is:
No Solution.
Explanation:
- Since parallel lines do not meet, there are no coordinates [tex]\((x, y)\)[/tex] that solve both equations at the same time.
