Question

The table below models the cost, y, of using a high-efficiency washing machine and a standard washing machine over x number of years.
Which equation represents the cost of the high-efficiency washing machine over a given number of years?

Which equation represents the cost of the standard washing machine over a given number of years?

After how many years of use would the washing machines cost the same amount?

Which washing machine would be the more practical purchase if kept for 9 years?

Answer 1

Answer:

Q1:  y = 25x + 500

Q2: y = 30x + 400

Q3: 20 years

Q4: Standard washing machine

Step-by-step explanation:

1 - First Problem

"Which equation represents the cost of the high-efficiency washing machine over a given number of years?"

NOTE: There will be a lot of reading at first, however, it covers the process of the first question and the second.

1.1 - Pinpointing key terms

This is a useful skill that never stops being practiced. The trick to it in this situation is to cross-examine information which, in our case, is the word problem and the graph.

Doing this we will notice this;

"Which equation represents the cost of the high-efficiency washing machine over a given number of years?"

For further clarification, we will look specifically at these two columns in our chart to answer the first problem

1.2 - Deciding which formula would best represent the data

The slope-intercept formula is used to measure data trends, and as such we can use it, and its rules, here. (Technically f(x) in general is universally recognized for this.)

1.3 - Determine which column to assign as the x values and which to assign as y values

Our y-axis is what we're measuring for, and that we want to find, and our x-axis is for the values that which our y values change in correlation to.

We want to find the high-efficiency cost, we'll as a result assign that column as the y values, at any given year, which in this case was given

1.4 - Find the slope

To do this we'll use the slope formula

$Slope = m = \frac{rise}{run} = \frac{change~in~y}{change~in~x} = \frac{y_2~-~y_1}{x_2~-~x_1}$

1.4.1 - Defining variables of slope formula

$x__1$ means the first x value of two consecutive points

$x__2$ second x value

(Must be to the right of the first point on a graph, greater than)$.^{*Note~1}$

NOTE: $y__{2}$ $is~derived~from~the~y~value~that~corresponds~with~x__2$; $and$ $y__1$ $from~x__1$

Further help in understanding this concept

To help you visualize I'll convert the variables to points

$\left[\begin{array}{ccccc}Number~of~years&|&high-efficiency~cost&|&point\\1&|&525&|&(1, 525)\\2&|&550&|&(2, 550)\\3&|&575&|&(3,575)\end{array}\right]$

1.4.2 -Putting it all together

In this case, we'll use (1, 525) and (2, 550)

As a result of defining of variables section, our x1 and y1 is (1, 525) and our x2 and y2 is (2, 550)

So, now lets plug and solve

(y2-y1)/(x2-x1)

On top:

y2 - y1 = 550-525 = 25

On the bottom

x2 - x1 = 2 - 1 = 1

Therefore our slope is 25/1 which is 25

1.5 - Finding our y-intercept, b, for y = mx + b

All we need to do now that we know the slope, and that we have a chart, is continue the trend in reverse to find y

because our slope was 25/1 that means that y increases by 25 for every increase of 1 that occurs in our x;

The reverse is for every decrease of 1 x, y decreases by 25

Using that we'll need to decrease our x and y accordingly until x equals 0

 (1, 525)

- (1, 25)

 (0, 500)

Our y intercept, b, in this case is 500

Therefore now we have the needed information to complete the formula

y = mx + b

y = 25x + 500

2 - Second Problem

"Which equation represents the cost of the standard washing machine over a given number of years?"

2.1-2 - Assigning x and y

y = Standard Cost

x = Number of years (given)

y = mx + b

2.3 - Find the slope

$Slope = m = \frac{rise}{run} = \frac{change~in~y}{change~in~x} = \frac{y_2~-~y_1}{x_2~-~x_1} = \frac{460~-~430}{2~-~1} = \frac{30}{1} = 30$

2.4 - Find the y-intercept, b, using the slope

 (1, 430)

- (1, 30)

(0, 400)

y-intercept = b = 400

2.5 - Plug it in slope-intercept form

y = mx + b

y = 30x + 400

3 - Third Problem

"After how many years of use would the washing machines cost the same amount?"

We need to set the two equations equal to each other, and thanks to the fact they're both equal to y, this allows us to do so through substitution

3.1 - Write the equations

y = 25x + 500

y = 30x + 400

3.2 - Substitute an equation in for the y of the other and simplify and solve algebraically

30x + 400 = 25x + 500

       - 400             -400

-25x            -25x            .

           5x = 100

             x = 20

After 20 years they will be equal

4 - Fourth Problem

"Which washing machine would be the more practical purchase if kept for 9 years? "

To answer this we just plug in 9 for x in each equation and see which is cheaper

4.1 - Rewrite

High Efficiency

y = 25x + 500

Standard

y = 30x + 400

4.2 - Plug in

High Efficiency

y = 25(9) + 500

Standard

y = 30(9) + 400

4.3 - Solve

High Efficiency

y = 25(9) + 500

y = 225 + 500

y = 725

Standard

y = 30(9) + 400

y = 270 + 400

y = 670

In this case the standard would cost less therefore our answer is the standard washing machine

Answer 2

ANSWER:

i) y = 25x + 500

ii) y = 30x + 400

iii) The washing machines would cost the same amount after 20 years of use

iv) Standard machine

Step-by-step explanation:

i)

We are to determine a straight line equation that models the cost of High-Efficiency washing machine over the years;

The first step step is to determine the slope of the line,

( change in y) / ( change in x ) = (550 - 525) / ( 2 - 1) = 25

The equation is slope-intercept form will be;

y = 25x + c

Where y is the cost of the High-Efficiency washing machine and x the number of years. To determine the y-intercept, c, we use any pair of points given in the data table;

when x = 1, y = 525

525 = 25(1) + c

c = 500

Therefore;

y = 25x + 500

ii)

The straight line equation that models the cost of Standard washing machine over the years;

Slope = (460 - 430) / (2 - 1) = 30

The equation is slope-intercept form will be;

y = 30x + c

when x = 1, y = 430

430 = 30(1) + c

c = 400

Therefore;

y = 30x + 400

Where y is the cost of the standard washing machine and x the number of years.

iii)

Given the cost functions for both machines over the number of years, we simply equate the two equations and determine the value of x when both machines would cost the same amount;

We have the cost functions;

y = 25x + 500

y = 30x + 400

Equating the two and solving for x;

25x + 500 = 30x + 400

500 - 400 = 30x - 25x

100 = 5x

x = 20

Therefore, the washing machines would cost the same amount after 20 years of use.

iv)

In order to determine which machine would be the more practical purchase if kept for 9 years we use the cost functions obtained in i) and ii)

The cost function of the High-Efficiency washing machine is;

y = 25x + 500

To determine the cost, we solve for y given x = 9

y = 25(9) + 500

y = 725

The cost function of the Standard washing machine is;

y = 30x + 400

We solve for y given x = 9

y = 30(9) + 400

y = 670

Comparing the two values obtained, the cost for the Standard washing machine is more practical.