Question
3. a) Graph the system x y 5 and 3x 4y 12 by hand.
b) Solve the system using your graph.
c) Verify your solution using graphing technology.
b) Solve the system using your graph.
c) Verify your solution using graphing technology.
Answers
GPT 3.5
a) To graph the system x + y = 5 and 3x + 4y = 12, we will first rewrite the equations in slope-intercept form.
For the first equation x + y = 5, we can solve for y:
y = 5 - x
For the second equation 3x + 4y = 12, we can solve for y:
4y = 12 - 3x
y = (12 - 3x)/4
b) To solve the system using the graph, we can plot the lines corresponding to the two equations and find their point of intersection.
Plotting the line y = 5 - x:
- Start with the y-intercept at (0, 5).
- Use the slope 1 and the intercept to find another point. Since the slope is 1, we can move 1 unit to the right and 1 unit down to get the point (1, 4).
- Connect these two points with a straight line.
Plotting the line y = (12 - 3x)/4:
- Start with the y-intercept at (0, 3).
- Use the slope -3/4 and the intercept to find another point. Since the slope is -3/4, we can move 4 units to the right and 3 units down to get the point (4, 0).
- Connect these two points with a straight line.
The lines intersect at the point (3, 2).
c) To verify the solution using graphing technology, we can use a graphing calculator or an online graphing tool to plot the lines and find their point of intersection.
By inputting the equations x + y = 5 and 3x + 4y = 12 into a graphing tool, we can see that the lines intersect at the point (3, 2). This confirms our solution from part b).
For the first equation x + y = 5, we can solve for y:
y = 5 - x
For the second equation 3x + 4y = 12, we can solve for y:
4y = 12 - 3x
y = (12 - 3x)/4
b) To solve the system using the graph, we can plot the lines corresponding to the two equations and find their point of intersection.
Plotting the line y = 5 - x:
- Start with the y-intercept at (0, 5).
- Use the slope 1 and the intercept to find another point. Since the slope is 1, we can move 1 unit to the right and 1 unit down to get the point (1, 4).
- Connect these two points with a straight line.
Plotting the line y = (12 - 3x)/4:
- Start with the y-intercept at (0, 3).
- Use the slope -3/4 and the intercept to find another point. Since the slope is -3/4, we can move 4 units to the right and 3 units down to get the point (4, 0).
- Connect these two points with a straight line.
The lines intersect at the point (3, 2).
c) To verify the solution using graphing technology, we can use a graphing calculator or an online graphing tool to plot the lines and find their point of intersection.
By inputting the equations x + y = 5 and 3x + 4y = 12 into a graphing tool, we can see that the lines intersect at the point (3, 2). This confirms our solution from part b).