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.
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).