Sunny G. answered • 08/01/20
Math Tutor Specializing in Elementary through High School Math
a) First, choose variables to represent the number of hours worked and the total amount money you would have earned. In this example we'll use "x" and "y" respectively. Job A states that it will pay you $12 per hour. That means that to find the amount of money that you have earned per hour, multiply 12 by the number of hours you've worked. This gives you 12x. Then the question states that job A will give you a $100 bonus at the end of the month. This means that at the end of the month, you will get $100 no matter how many hours you have worked. Therefore, you can add 100 to 12x to calculate the total amount you would have earned at the end of the month based on the number of hours you have worked.
For job B, they are saying that it will pay you $16 per hour, with no added bonus (aka a $0 bonus). Do the same thing you did for job A to find the amount money that you have earned per hour for job B by multiplying 16 by x, giving you 16x. Since theres no bonus for job B, you don't need to add anything. These calculation will give you the following equations:
Job A: y = 12x + 100
Job B: y = 16x
b) You can put both of these equations into a graphing calculator but if you can't use one then I will tell you how to graph it by hand.
First lets graph job A. Start with plotting the y-intercept, which is the point on the graph where x equals 0. In this case the y-intercept is 100 because when you plug "0" into the equation, you get 100. Next plot the slope, which is the number multiplying the "x". reminder: the formula for slope is m = rise divided by run. This means that the slope for job A's graph is 12 divided by 1, meaning that from any point on the graph, if you go 12 units up and 1 unit over to the right, you will get another point on the graph. All of the points on the graph will follow this ratio. Start from the y-intercept, follow the slope ratio, and connect the dots to graph the equation.
For job B, do the same thing you did with job A's graph, except this time for the y-intercept use "0" and for the slope use m = 16.
c) The intersection point is the point where the x and y values are the same for both graphs. Since the lines are not parallel, they will inevitably intersect somewhere. You can find the intersection point by using a graphing calculator but if you can't use one again then I will show you how an easy way to calculate it. Since y is common and isolated for both equations, you can set them equal to each other. This will give you
12x + 100 = 16x. Solve for x by subtracting 12x on both sides, giving you 100 = 4x, and then divide both sides by 4, giving you x = 25. Plug 25 into one of the equations to get y. when you plug 25 for x into job A's equation, you get (12)(25) + 100 = 400. To check you can plug 25 for x into job B's equation and see if they give you the same answer. The intersection point is (25, 400). In context this means that the point where you make the same amount of money at each job ($400) is when you work 25 hours.
d) The "most" amount of money depends on the number of hours you work. Use the intersection point to answer this. An easy way to figure this out using math is to take a number smaller than the x value of the intersection point, plug it into both equations, and compare the y values they give you. For example, let's use x = 10. When you plug it in, job A will give you y = 12(10) + 100 which gives you y = 220. job B will give you y = 16(10) which gives you y = 160. Using this test, you can conclude that if you want to work for less that 25 hours, you will make more money at Job A, and if you want to work for more than 25 hours, you will make more money at Job B.
You can also answer this by simply looking at the graph. Whichever equation have a higher y-value for the number of hours you want to work is the one you should choose to make more money. Your choice will differ before and after the intersection point.
