Once you have an equation to describe a linear relation, you can use it to solve for other terms. To solve for the y value, substitute the value for x and solve. To solve for the x value, substitute the
value for y and isolate to get x.
Consider the following relationship between the number of toppings on a pizza and the cost:
| # Toppings |
0 |
1 |
2 |
3 |
| Cost ($)
| 10.00 |
11.50 |
13.00 |
14.50 |
Since I love toppings, I want to know how much a pizza would cost if I had 15 toppings. One option is to continue the pattern. The pattern starts at 10 and increases by 1.50 each time.
This could take a long time. Instead, we can write an equation and solve the equation for 15 toppings:
\(y=mx+b\)
\(y=1.5x+10\)
Here, x represents the number of toppings and y represents the cost. We want to know the cost when \(x=15\). Plug in and solve:
\(y=1.5x+10\)
\(y=1.5(15)+10\)
\(y=22.50+10\)
\(y=32.50\)
Therefore, the cost of a pizza with \(15\) toppings is \(y = $32.50\).
We can also use the equation to solve for the number of toppings given the price. How many toppings are on the pizza if the price of the pizza was \($19\).
We want to know the number of toppings when \(y=19\). Plug in and isolate for x:
\(y=1.5x+10\)
\(19=1.5x+10\)
\(19-10=1.5x\)
\(9=1.5x\)
\(9/1.5=x\)
\(6=x\)
Therefore, there were \(6\) toppings if the pizza cost \(y = $19.00\).
Given the table of values for x and y, solve for when x = 40.
| X |
1 |
2 |
3 |
4 |
5 |
6 |
7 |
| Y
| 2 |
5 |
8 |
11 |
14 |
17 |
20 |
Show Answer
First, calculate the slope. The value of y increases by 3 each time. The value of x increase by 1. Therefore, the slope is 3.
Now find the y intercept (when \(x=0\)). In this case it would be \(2 - 3 = -1\) working backwards from \(x=1\).
The equation of the line is
\( y = 3 x - 1\)
Now, plug in \(x=40\) and solve for y:
\( y = 3 (40) - 1 = 119\)
Using the same table as the question above, what would be the x value when y = 40?
Show Answer
Since the table remains the same, the slope, intercepts and equations also stays the same:
\( y = 3 x - 1\)
Plug in \(y=40\) and solve for x:
\( 40 = 3 x - 1\)
\( 3 x - 1 + 1 = 40 + 1\)
\( x = \cfrac{41}{3}\)
Gabriel's internet bill is $30/ month plus $10/GB for every data over the limit (25 GB). Write a linear equation to represent Gabriel's bill.
Show Answer
The dependent variable is the internet bill cost (C) which changes based on the amount of data (d) over the limit. The amount of data (GB) over the limit would be the independent variable.
Since the problem states the base payment is $30/month, that would be the starting value or the y-intercept. This is the price of the bill when Gabriel is 0 GB over the limit.
Next, the question states that for every GB over the limit, an additional $10 would be charged. That means the bill will increase $10 for each GB over the limit. This value is the slope.
The equation is:
\(C = 10(d) + 30\)