Every straight line that is not vertical will cut the Y axis at a single point. The y coordinate of this point is called y intercept of the line.
A line with slope m and y intercept b can be expressed through the equation
y = mx + b
When the slope (m) and y-intercept (b) of a line are given, we can use slope intercept form to find the equation of a line.
∙ If a line with slope m, makes x intercept d, then the equation of the straight line is x = my + d.
∙ y = mx is the equation of a straight line passing through the origin.
Example 1 :
Find the slope and y-intercept of the straight line whose equation is 4x - 2y + 1 = 0.
Solution :
Write the given equation 4x - 2y + 1 = 0 in slope-intercept form.
4x - 2y + 1 = 0
4x + 1 = 2y
Divide each side by 2.
(4x + 1)/2 = y
2x + 1/2 = y
or
y = 2x + 1/2
The above form is slope intercept form.
If we compare y = 2x + 1/2 and y = mx + b, we get
m = 2 and b = 1/2
Problem 2 :
A straight line has the slope 5. If the line cuts y-axis at -2, find the general equation of the straight line.
Solution :
Because the line cuts y-axis at -2, clearly y-intercept is -2.
Now, we know that slope m = 5 and y-intercept b = -2.
Equation of a straight line in slope-intercept form is
y = mx + b
Substitute 5 for m and -2 for b.
y = 5x - 2
Subtract y from each side.
0 = 5x - y - 2
General equation of the required line :
5x - y - 2 = 0
Problem 3 :
Find the equation of the line in slope-intercept form.
Solution :
The above line is a falling line. So, its slope will be a negative value.
Measure the rise and run.
From the above diagram, rise = 4 and run = 3.
slope = rise/run
= -4/3
The line in the diagram above intersects y-axis at -1.
So, y-intercept is -1.
Equation of the line in slope intercept form :
y = mx + b
Substitute -4/3 for m and -1 for b.
y = -4x/3 - 1
One of the major applications of slope intercept form equation of a line is, it can be used as linear cost function in business.
Linear cost function is the function where the cost curve of a particular product will be a straight line.
Mostly this function is used to find the total cost of 'n' units of the products produced.
For any product, if the cost curve is linear, the linear cost function of the product will be in the form of
y = Ax + B
Here,
'y' stands for total cost
'x' stands for number of units.
'A' stands for cost of one unit of the product.
'B' stands for fixed cost.
Linear cost function is a bi parametric function, as there are two parameters 'A' and 'B'.
Once the two parameters 'A' and 'B' are known, the complete function can be known.
The following steps are involved in solving word problems on linear cost function.
Step 1 :
First we have to go through the question carefully and understand the information given in the question.
After having gone through the question, we have to conclude whether the information given in the question fits linear-cost function.
If the information fits the linear-cost function, we have to follow step 2
Step 2 :
Target :
We have to know what has to be found.
In linear-cost function, mostly the target would be to find either the value of 'y' (total cost) or 'x' (number of units).
Step 3 :
In step 3, we have to calculate the two constants "A" and "B" from the information given in the questions. It has been shown clearly in the example problem given below.
Step 4 :
Once the values of 'A' and 'B' in y = Ax + B are found, the linear-cost function would be completely known.
Step 5 :
After step 4, based on the target of the question, we have to find either the value of 'y' or 'x' for the given input.
For example, if the value of 'x' (number of units) is given, we can find the value of 'y' (total cost).
If the value of 'y' (total cost) is given, we can find the value of 'x' (number of units).
Problem :
A manufacturer produces 80 units of a particular product at a cost of $ 220000 and 125 units at a cost of $ 287500. Assuming the cost curve to be linear, find the cost of 95 units.
Solution :
Step 1 :
When we go through the question, it is very clear that the cost curve is linear.
And the function which best fits the given information will be a linear-cost function.
y = Ax + B
y ----> Total cost
x ----> Number of units
Step 2 :
Target :
We have to find the value of 'y' for x = 95.
Step 3 :
From the question, we have
x = 80 and y = 220000
x = 75 and y = 287500
Step 4 :
When we substitute the above values of 'x' and 'y' in
y = Ax + B,
we get
220000 = 80A + B
287500 = 75A + B
Step 5 :
When we solve the above two linear equations for A and B, we get
A = 1500, B = 100000
Step 6 :
From A = 1500 and B = 100000, the linear-cost function for the given information is
y = 1500x + 100000
Step 7 :
To estimate the value of 'y' for x = 95, we have to substitute 95 for x in
y = 1500x + 100000
y = 1500 x 95 + 100000
= 142500 + 100000
= 242500
So, the cost of 95 units is $242500.
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