Due Tuesday 9/5 via Teams by 2:00 pm
This is a brief math exercise to help you practice a few key techniques we'll use during the semester. Please write out the answers below on a separate sheet of paper and then scan and submit your answer via Teams. Also, please show all your work and note that the phrase solve algebraically means to use algebra, not to find the solution by plotting points on graph paper.
1. Please graph the two equations below and solve algebraically for the (x,y) point where they intersect. Be sure to label the axes, show the numerical value of each intercept, and indicate the slope of each line.
`y = 200 - 10 x`
`x + 1 = 0.05 y`
2. On your graph from part 1 draw a horizontal line through the intersection of the two equations. Then calculate the area of the triangle above the line you just drew, below the `y=200-10 x` equation, and to the right of the y-axis.
3. Graph the three equations below and solve algebraically for the coordinates of the three points where pairs of lines intersect. Be sure to label the graph thoroughly. Then calculate the area of the triangle formed by the points.
`y = 1000 - 10 x`
`y = 200 + 10 x`
`x = 20`
4. Variables `x_1` and `x_2` both depend on `y` as shown by the equations below. Please solve algebraically for the equation of a new variable, `x_3`, which also depends on `y` and is equal to the sum of `x_1` and `x_2`. Plot the three equations on separate graphs (`y` on the vertical axis and `x_1`, `x_2` or `x_3` on the horizontal axis) and show the intercepts of each one.
`x_1 = 10 - y`
`x_2 = 20 - 2*y`
5. Finally, add the line `y=5` to each of the graphs you drew for part 4. Then calculate the areas of the three triangles that lie below the lines from part 4, above the `y=5` line, and to the right of the y- axis. What do you notice about the area of the third triangle compared to the areas of the other two?