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
[tex]Slope = m = \frac{rise}{run} = \frac{change~in~y}{change~in~x} = \frac{y_2~-~y_1}{x_2~-~x_1}[/tex]
1.4.1 - Defining variables of slope formula
[tex]x__1[/tex] means the first x value of two consecutive points
[tex]x__2[/tex] second x value
(Must be to the right of the first point on a graph, greater than)[tex].^{*Note~1}[/tex]
NOTE: [tex]y__{2}[/tex] [tex]is~derived~from~the~y~value~that~corresponds~with~x__2[/tex]; [tex]and[/tex] [tex]y__1[/tex] [tex]from~x__1[/tex]
Further help in understanding this concept
To help you visualize I'll convert the variables to points
[tex]\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][/tex]
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
[tex]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[/tex]
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
