Bruce drives his car for his job. The equation R=0.575 m+42...

Bruce drives his car for his job. The equation $R=0.575 m+42$ models the relation between the amount in dollars, $R,$ that he is reimbursed and the number of miles, $m,$ he drives in one day. (a) Find the amount Bruce is reimbursed on a day when he drives 0 miles. (b) Find the amount Bruce is reimbursed on a day when he drives 220 miles. (c) Interpret the slope and $R$ -intercept of the equation. (d) Graph the equation.

Bruce drives his car for his job. The equation $R=0.575 m+42$ models the relation between the amount in dollars, $R,$ that he is reimbursed and the number of miles, $m,$ he drives in one day.
(a) Find the amount Bruce is reimbursed on a day when he drives 0 miles.
(b) Find the amount Bruce is reimbursed on a day when he drives 220 miles.
(c) Interpret the slope and $R$ -intercept of the equation.
(d) Graph the equation.

Key Concepts

Linear Functions

A linear function models a relationship between two variables where the change is consistent across the domain. It is defined by a constant rate of change (the slope) and a fixed starting value (the intercept), making it useful for representing real-world relationships such as cost, distance, or time where one variable changes at a steady rate with respect to another.

Y-Intercept

The y-intercept of a linear equation is the value of the dependent variable when the independent variable is zero. It represents the starting value or initial condition in the context of a real-world problem, indicating the amount present without any contribution from the changing factor.

Slope

The slope in a linear equation indicates the rate at which the dependent variable changes with respect to the independent variable. It quantifies the effect of a one unit increase in the independent variable on the dependent variable, providing insight into proportional changes within the modeled scenario.

Graphing Linear Equations

Graphing linear equations involves plotting the line represented by the equation on a coordinate plane using the slope and y-intercept. This visual representation is key for understanding the relationship between variables as it shows both the constant rate of change and the starting point, making it easier to predict values and interpret the behavior of the function.

Transcript

00:01 The first thing we're going to do to solve this a problem is to relate this equation to the y intercepts, the slope intercept form, the slope intercept form of a line, y plus mx plus b.

00:18 So we want to rearrange ours to look like this.

00:23 Our r is going to be the y.

00:26 Our m is the 0 .575.

00:32 The x is the m, and so this m, and then the b is the 42.

00:42 Now we have our equation in slope intercept form.

00:46 So instead of making an x, y chart to investigate this, we're going to make an mr chart.

00:52 Okay, also for our graph, we are going to let our x axis be miles driven in a day, and the y axis.

01:03 Be the reimbursement for the day.

01:10 Okay, so if the car was driven zero miles, then we would have 0 .575, sorry about that, 0 .575 times 0 plus 42.

01:31 So that would be 0 plus 42 is equal to our reimbursement.

01:36 So our reimbursement for zero miles would be $42.

01:43 Okay, let's say that the car was driven 220 miles in a day.

01:48 We're going to let mb 220, so we'll have the reimbursement equals 0 .575 times 220 miles driven in the day, plus 42.

02:04 So our reimbursement would be 5 .575 times 220 is 126 .5 plus the $42 just for driving the car.

02:21 So the total reimbursement for that day would be $168 .50.

02:36 Okay...

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