Use the equation for cost to rent a backhoe in Exercise 24 to answer the following: a. If the sl

Use the equation for cost to rent a backhoe in Exercise 24...

Key Concepts

Linear Equations

A linear equation is an equation that can be graphically represented as a straight line. It is typically written in the form y = mx + b, where 'm' is the slope that indicates how much the dependent variable changes per unit change in the independent variable, and 'b' is the y-intercept, which represents the value of y when the independent variable is zero.

Slope

The slope is the coefficient of the independent variable in a linear equation. It defines the rate of change of the dependent variable relative to changes in the independent variable. Altering the slope changes the steepness of the line, meaning it affects how quickly the values of the dependent variable increase or decrease as the independent variable increases.

Vertical Intercept (y-intercept)

The vertical intercept, or y-intercept, is the value of the dependent variable when the independent variable is zero. It is represented by 'b' in the equation y = mx + b and typically stands for the base value or fixed cost before any changes in the independent variable occur.

Horizontal Intercept

The horizontal intercept is the point where the graph crosses the x-axis, which occurs when the value of the dependent variable is zero. It is calculated by setting y = 0 in the equation and solving for the independent variable. This intercept can provide insights into break-even points or thresholds within a given context.

Transcript

00:01 In the given question, we are told that we have to look at an equation that is given to us in the question and answer some questions regarding this equation.

00:13 The equation is represented by, it is given by the situation that the cost in dollars to rent a backhoe for h hours is represented as c is equal to 85h plus 55.

00:38 So, what you can see over here is c is the cost in dollars to rent a backhoe for h hours.

01:04 So, we can just denote h over here as the time that is taken in hours.

01:17 So, in this equation, what we are asked about in the question is that if the slope of this equation is changed to 100, how would that affect the vertical intercept of the graph? so, before going into the vertical intercept, let's just look at this equation that we have at hand.

01:44 So, there is a form, a certain form of equation called the slope intercept form of a line.

01:52 Slope intercept form, slope intercept form of a line and this kind of an equation is represented of the form y equals m times x plus c.

02:10 And, what we could see in these types of equations is that m over here gives us the slope of the graph and c gives us the vertical intercept or the y intercept, vertical intercept.

02:31 So, if you know the value of c, you can write the vertical intercept as 0c.

02:40 So, you would ask what is the vertical intercept, right? so, the vertical intercept is nothing but the point, the point where the graph intersects the vertical axis or we can call it the y -axis.

03:10 And, when we are taking a point on the y -axis, you should know that any point on the y -axis would have its x -coordinate as 0, right? so, that is why we have the y vertical intercept as a form, in the form of 0c.

03:33 So, this is exactly what we usually do to find the y -intercept also.

03:39 What we do is to find the vertical intercept, we just substitute, substitute the value of the x -coordinate, the value of the x -coordinate as 0, x -coordinate, x -coordinate as 0.

04:03 So, now that we know all this about the type of equation that is given to us in the question, what we could do next is we can take the, we can just find the vertical intercept and the slope of the equation that is given to us in the question, right? so, the equation that is given to us is of the form c is equal to 85h plus 55.

04:37 So, over here the slope that is denoted as m would be 85, right? and, the vertical intercept would be at the point, vertical intercept would be at the point 0, 55.

05:00 So, now we can write, if the slope, if the slope is changed to 100, is changed to 100, 100, c is then equal to 100h plus 55, right? so, now we can write here the slope is equal to 100 and the vertical intercept, the slope is equal to 100 and the vertical intercept, vertical intercept has no change, right? vertical intercept has no change, no change, it is still at, it is still at the point 0, 55.

06:02 So, this is what we can answer for the first part of the question, that is even when the slope of this graph is changed, the vertical intercept of this graph does not change.

06:16 So, now let's move on to the second part of this question, where we are asked, if the slope of the equation is changed to 100, how would that affect the horizontal intercept of the graph? so, when you take the horizontal intercept, what you could see is that the horizontal intercept, the horizontal intercept is the point at, the point at which, at which the graph intersect the horizontal axis.

07:00 The graph intersect, intersect the horizontal axis, which is the x -axis, right? and what we could see in the case of horizontal intercept is that, just as in the case of the vertical intercept, we substituted the value of x -coordinate as 0, what you do to find the horizontal intercept is to substitute the value, substitute, substitute the value of, the value of the y -coordinate as 0...

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