Certainly! Let's solve the given systems of equations using the graphing method step-by-step.
### System A
[tex]\( y = x + 2 \)[/tex] and [tex]\( y = -x + 8 \)[/tex]
1. Graph the first equation: [tex]\( y = x + 2 \)[/tex]
- This is a linear equation in slope-intercept form ([tex]\( y = mx + b \)[/tex]), where [tex]\( m \)[/tex] (the slope) is 1 and [tex]\( b \)[/tex] (the y-intercept) is 2.
- To plot the graph, start at the y-intercept (0, 2).
- Use the slope 1 (which means rise 1, run 1) to find another point: from (0, 2), move up 1 and right 1 to (1, 3).
- Draw the line through these points.
2. Graph the second equation: [tex]\( y = -x + 8 \)[/tex]
- This is also a linear equation in slope-intercept form ([tex]\( y = mx + b \)[/tex]), where [tex]\( m \)[/tex] (the slope) is -1 and [tex]\( b \)[/tex] (the y-intercept) is 8.
- To plot the graph, start at the y-intercept (0, 8).
- Use the slope -1 (which means rise -1, run 1) to find another point: from (0, 8), move down 1 and right 1 to (1, 7).
- Draw the line through these points.
3. Find the intersection of the two lines
- The point where both lines intersect is the solution to the system.
- By visually inspecting or solving, we find that the lines intersect at [tex]\( (3, 5) \)[/tex].
Thus, the solution to System A is [tex]\( (3, 5) \)[/tex].
### System B
[tex]\( y = 2x - 5 \)[/tex] and [tex]\( y = -3x + 5 \)[/tex]
1. Graph the first equation: [tex]\( y = 2x - 5 \)[/tex]
- This is a linear equation in slope-intercept form ([tex]\( y = mx + b \)[/tex]), where [tex]\( m \)[/tex] (the slope) is 2 and [tex]\( b \)[/tex] (the y-intercept) is -5.
- To plot the graph, start at the y-intercept (0, -5).
- Use the slope 2 (which means rise 2, run 1) to find another point: from (0, -5), move up 2 and right 1 to (1, -3).
- Draw the line through these points.
2. Graph the second equation: [tex]\( y = -3x + 5 \)[/tex]
- This is also a linear equation in slope-intercept form ([tex]\( y = mx + b \)[/tex]), where [tex]\( m \)[/tex] (the slope) is -3 and [tex]\( b \)[/tex] (the y-intercept) is 5.
- To plot the graph, start at the y-intercept (0, 5).
- Use the slope -3 (which means rise -3, run 1) to find another point: from (0, 5), move down 3 and right 1 to (1, 2).
- Draw the line through these points.
3. Find the intersection of the two lines
- The point where both lines intersect is the solution to the system.
- By visually inspecting or solving, we find that the lines intersect at [tex]\( (2, -1) \)[/tex].
Thus, the solution to System B is [tex]\( (2, -1) \)[/tex].
In summary:
- The solution to System A is [tex]\( (3, 5) \)[/tex].
- The solution to System B is [tex]\( (2, -1) \)[/tex].
