a company finds it can produce 5 heaters for $1750, while producing 10 heaters costs $2900. express the cost y, as a linear function of the number of heaters x.
how do u begin 2 do this??
y=mx+b
1750=5m+b
2900=10m+b
solve for m and b. To get m, subtract one equation from the other to get:
-1250=-5m
then to get b, put it back in either equation.
What is a linear function?
To express the cost (y) as a linear function of the number of heaters (x), you can use the slope-intercept form of a linear equation: y = mx + b.
1. Identify two points on the line:
We have two points: (5, 1750) and (10, 2900). These points represent the number of heaters produced and their respective costs.
2. Find the slope (m):
The slope (m) represents the change in cost for each additional unit of production. It can be calculated using the formula: m = (y2 - y1) / (x2 - x1).
Let's use the points (5, 1750) and (10, 2900) to calculate the slope:
m = (2900 - 1750) / (10 - 5)
= 1150 / 5
= 230
The slope, in this case, is 230.
3. Substitute one point and the slope into the equation:
By substituting the slope and one point (x1, y1) into the slope-intercept form (y = mx + b), we can solve for the y-intercept (b). You can choose either of the two points given.
Let's use the point (5, 1750) to substitute into the equation:
1750 = 230(5) + b
4. Solve for b:
Simplify the equation by multiplying the slope and the x-coordinate, and then isolate b:
1750 = 1150 + b
b = 1750 - 1150
b = 600
5. Write the linear function:
Now that we have the slope (m = 230) and the y-intercept (b = 600), we can write the linear function in slope-intercept form:
y = 230x + 600
Therefore, the cost (y) can be expressed as a linear function of the number of heaters (x) using the equation y = 230x + 600.
To express the cost y as a linear function of the number of heaters x, you need to find the equation of the straight line that represents the relationship between the cost and the number of heaters.
Here, you have two data points: (5, $1750) and (10, $2900). The number of heaters, x, will be our independent variable, and the cost, y, will be our dependent variable.
To find the equation, we can use the slope-intercept form: y = mx + b, where m represents the slope and b represents the y-intercept.
Step 1: Find the slope (m):
The slope of a line can be calculated using the formula m = (y2 - y1) / (x2 - x1) where (x1, y1) and (x2, y2) are any two points on the line.
Using the two data points, (5, $1750) and (10, $2900), we have:
m = (2900 - 1750) / (10 - 5) = 1150 / 5 = 230
So, the slope of the line is 230.
Step 2: Find the y-intercept (b):
To find the y-intercept, substitute the values of one of the points into the equation y = mx + b and solve for b.
Using the point (5, $1750):
1750 = 230(5) + b
1750 = 1150 + b
b = 1750 - 1150
b = 600
So, the y-intercept of the line is 600.
Step 3: Write the linear equation:
Now, we have the slope (m = 230) and the y-intercept (b = 600), so we can write the equation of the line:
y = 230x + 600
This is the linear function that expresses the cost (y) as a function of the number of heaters (x).
Therefore, the cost y, as a linear function of the number of heaters x, is given by the equation y = 230x + 600.