To answer this question, we need to know that this is a problem of a linear function. If we see the problem, we have:
1. Mary is spending money at an average rate of $5 per day. In this case, it represents the slope of the line. The slope of the line is the rate of change of this function. Then, we have that the slope is m = -5 --$5 per day (because she is spending money. She is going to have less and less money as time passes.) The slope is given by the difference in the y-coordinates over the difference in the x-coordinates, or, in this case, change in money (y variable), and change in days (x variable). We are told that the amount left depends on the number of days (money is a dependent variable of time):
[tex]m=\frac{y_2-y_1}{x_2-x_1}[/tex]2. In the question, we are given a point of the function (14 days, $68) (see below in the graph.)
Therefore, we have the slope of the line, and also a point of the line. As a result, we can write the equation in the point-slope form of this situation as follows:
[tex]y-y_1=m(x-x_1)[/tex]We have that:
m = -5
x1 = 14
y1 = 68
Then, we have the point-slope form for the situation:
[tex]y-68=-5(x-14)[/tex]And to find the slope-intercept form of the line, we need to simplify the previous expression as follows:
[tex]y=mx+b[/tex]Then, we have:
1. First, apply the distributive property:
[tex]y-68=-5(x-14)\Rightarrow y-68=-5x+70[/tex]2. Now, add 68 to both sides of the equation:
[tex]y-68+68=-5x+70+68\Rightarrow y=-5x+138[/tex]The graph for this line is:
We can see the graph with the values of the x- and y-intercepts. In the beginning, Mary had $138, and after almost 28 days, at this rate, she will have no money with her.
In summary, we have that:
1. The equation in point-slope form is:
[tex]y-68=-5(x-14)[/tex]2. The equation in slope-intercept form is:
[tex]y=-5x+138[/tex]
