(a) In linear models, the x-variable is usually the independent variable, while the y-variable is the dependent variable. In this problem, we know that Paul's earnings depend on how many hours of additional yard work he does. Hence, the x-variable represents the number of hours Paul does extra yard work and the y-variable represents his earnings.

(b) A linear equation in point-slope form is,

$y-y_1=m(x-x_1)$

Where m is the slope and (x₁, y₁) is a point on the line.

In this case, we know that Paul earns an extra $1.50 per hour for additional yard work. If the additional yard work time in hours is represented by the x-variable, then the slope is 1.50.

Now, we need a point on the line. This is given in the problem, where it is said that when Paul worked 6 hours, he earned a total of $24. So the point is (6, 24),

$y-24=1.50(x-6)$

Hence, the equation in point-slope form is y - 24 = 1.50(x - 6).

(c) The slope-intercept form of the equation of a line is,

$y=mx+b$

We have to take the equation found in part (b) and rearrange it to find the equation in this form.

To do so, apply the distributive property to 1.50 on the right side of the equation,

$\begin{gathered} y-24=1.50x-1.50\cdot6 \\ \\ y-24=1.50x-9 \end{gathered}$

And add 24 to both sides,

$\begin{gathered} y-24+24=1.50x-9+24 \\ \\ y=1.50x+15 \end{gathered}$

The value of b is 15. Remember that b is the y-intercept, which is the value of the function when x = 0. This means that y = 15 when x = 0 and, therefore, 15 is the flat rate.

Hence, the equation in slope-intercept form is y = 1.50x + 15, and the y-intercept, b, represents the flat rate Paul earns without doing any extra work.

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1 year ago