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Slope intercept form,In mathematics, is a way to express the equation of a line. It's also the most frequently used way to express the equation of a line. When a straight line intersects a segment on the axis, it is called an intercept.
To be able to use slope intercept form, you'll need ‘slope of the line’ and ‘y- intercept of the line’. The variable 'm' represents the slope of a straight line and is utilised in the equation
y = mx+b.
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Key Takeaways- Slope intercept form, straight line, x-intercept, y-intercept, slope-intercept, slope
Also read- Definite Integral Formula
Slope Intercept Form Formula
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The formula for slope-intercept form is y = mx + b, here-
- m represents the slope of the line.
- b represents the y-value of the line’s y-intercept.
- x, y represent every point on the line.
The equation of a vertical line cannot be found through the application of the slope-intercept formula.
Slope intercept form formula
Also read- Planes
Slope Intercept Form of an Equation
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The formula for slope-intercept form is y = mx + b. Let us see how to derive the slope-intercept form equation of a straight line.
Step 1: Let L be a line with slope m and y-intercept b. Highlight the point that should be on the line. Justify your choice.
(b, 0) (0, b) (0, m) (m, 0)
The coordinate of x is 0 in the point that includes the y-intercept.
Step 2: Recall that slope is the ratio of change in y to change in x. Complete the equation for the slope m of the line using the y-intercept (0, b), and another point (x, y) on the line.
Slope m = change in y-values / change in x-values
Slope m = (y - b) / (x - 0)
Slope m = (y - b) / x
Step 3: Mostly y is written on one side of the equationIn an equation of a line. Solve the equation from Step 2 for y.
m = (y - b) / x
Multiply both sides by x
m.x = [(y - b) / x].x
mx = y - b
Add b to both sides of the equation.
mx + b = (y - b) + b
mx + b = y
Write the equation with y on the left side.
y = mx + b
Also read- Important formula
Slope Intercept Form Graph
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We obtain a straight line when we draw the graph for the slope-intercept form equation. The optimum form is slope-intercept. It is simple to graph or answer word problems based on it since it is in the form "y=". All we have to do now is plug in the x-values, and the equation for y is solved.
The best thing about the slope-intercept form is that we can get the slope and intercept values right out of the equation.
Also read- Algebra Formula
Slope Intercept Form x Intercept
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We can use the slope-intercept form (y = mx + b). here m and b are constants.
m = slope of the line
b = x- intercept of line
Setting y = 0 will give us the x-value at which the line crosses the x-axis, which is the x-intercept.
The slope of a line can also be expressed as tangent angle such as
m = tan θ
Slope Intercept Form Derivation
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The slope intercept form can be derived from the straight line equation in standard from as given below:
The straight line standard form of an equation can be written as:
Ax + By + C = 0
The equation can be rearranged as: By = -Ax - C
y = (-A/B)x + (-C/B)
This is of the form y = mx + c
Here the slope of the line is represented by (-A/B) and (-C/B) is the y-intercept.
Things to remember
- Straight lines are produced by linear functions.
- The slope-intercept formula is used when we know the slope of the straight line.
- For a line passing through the origin, the y-intercept will be (b = 0), so the equation will be of the form: y = mx.
- The slope-intercept formula cannot be applied to find the equation of a vertical line.
Also read: Isosceles Triangle Theorems
Sample Question
Ques- Using the slope intercept form, find the equation of a straight line with slope 1/3 and whose y-intercept is (0, -5). [2 marks]
Ans: To find the equation of the given line:
Given: the slope of the line is m = 1/3.
the y-intercept of the line is (0, b) = (0, -5) ⇒ b = -5.
Using the slope-intercept formula, the equation of the given line is,
y = mx + b
y = (1/3) x - 5
The equation of the given line is, y = (1/3) x - 5.
Ques- What is the y-intercept for the following equation:y + 84x = 157x + 250 [2 marks]
Ans: The y-intercept for an equation can be found through two ways. You could substitute 0 in for x.
This would give you:y + 84 ∗ 0= 157 ∗ 0 + 250.
you get: y = 250
However, this can also be done by finding the slope-intercept form of the line. You do this by solving for y. Indeed, this is very, very easy. Recall that the slope intercept form is:
Y = mx + b
This means that, as written, your equation obviously has b = 250.
Ques- Find the slope and y-intercept of the straight-line 4x - 7y + 1 = 0. [2 marks]
Ans: The equation of the given straight line is
4x - 7y + 1 = 0
⇒ 7y = 4x + 1
⇒ y = 4/7x + 1/7
Now, compare the above equation with the equation y = mx + b we get,
m = 4/7 and b =1/7.
Therefore, the slope of the given straight line is 4/7 and its y-intercept = 1/7 units.
Ques- With the slope 6 and y-intercept 4, find the equation of a line? [2 marks]
Ans: Here,
m = 6 and
b = 4
With the formula: y = mx + b
The equation of the line will be y = 6x + 4
Ques- Write the equation of the line in slope-intercept form where slope is -2 and passing through the point (0, 9). [2 marks]
Ans: Given,
Slope = m = -2
Point = (x, y) = (0, 9)
Equation of a line in slope-intercept form is:
y = mx + b….(i)
According to the given,
9 = (-2)(0) + b
b = 9
Substituting the value of m and b in equation (i),
y = -2x + 9
This is the required equation of a line in slope-intercept form.
Ques- Find the slope and y-intercept of the equation of line 2x – y + 5 = 0. [2 marks]
Ans: Given equation of a line:
2x – y + 5 = 0
Thus,
y = 2x + 5
This is of the form y = mx + b
Here, m = 2 and b = 5
Therefore, slope = 2 and y-intercept = 5
Ques- What is the y-intercept of the line with the following equation 2y− 4x = 10x − 20 [3 marks]
Ans:There are two ways that you can find the y-intercept for an equation. You could substitute 0 in for x. This would give you: 2y − 4∗ 0 = 10∗ 0 − 20
Simplifying, you get:
2y = -20
y = -10
Another way to do this is by finding the slope-intercept form of the line. You do this by solving for
2y = 14x − 20
Just divide everything by 2
y = 7x - 10
Remember that the slope-intercept form gives you the intercept as the final constant. Hence, it is -10
Ques- Find the equation of the line whose slope is 8 and the coordinates of the point are (3, 5). [2 marks]
Ans: Given, m = 8
(x1, y1) = (3, 5)
The formula for the equation of a line is,
y – y1 = m (x – x1)
y – 5 = 8 (x – 3)
y – 5 = 8x – 24
8x – y = 19
8x – y – 19 = 0
Ques- Form the equation of the straight line that runs parallel to another straight line of equation 2x + 3y + 11 = 0. The sum of the intercepts intercepted by the axis is 15. [3 marks]
Ans: If 2x + 3y + 11 is the equation of the line, then the equation parallel to this line will be 2x + 3y + c = 0
According to the question, a + b = 15 if the equation x/a + y/b =1 is considered, where a and b are intercepts cut on x as well as y axis.
Let’s rearrange 2x + 3y + c = 0 to bring it in x/a + y/b = 1 form.
2x + 3y = -c
x /(-c/2) + y/(-c/3) = 1
Hence, a = -c/2 and b = -c/3
Therefore,
-c/2 + (-c/3) = 15
-3c/6 - 2c/6 = 15
-5c/6 = 15
c = -18
So, the equation of the line is: 2x + 3y - 18 = 0
Ques- What are the coordinates of the foot of the perpendicular to the point with coordinate (-1,3) to the line with equation 3x - 4y - 16 = 0. [3 marks]
Ans: Let us assume that (a,b) are the coordinates of the foot of the perpendicular from the point (-1,3) to the line with equation 3x - 4y - 16 = 0.
Slope of the line (m1) is (b-3) / (a+1)
Slope of the line (m2) is ¾
As these lines are perpendicular, m1.m2 = -1
{(b-3) / (a+1)} . (¾) = -1
=> 4a + 3b = 5 …(1)
(a,b) lies on 3x - 4y = 16
Therefore, 3a - 4b = 16 …(2)
a = 68/25
b = -49/25
Thus, the coordinates are {68/25 , 49/25}
Ques- The equation of a given line is 3x - 7y + 1. Calculate the slope. [2 marks]
Ans: According to formula, we know that,
y - y1 = m (x - x1)
Therefore, x1 = 3, and y1 = -7
Slope of the line = -(3/-7)
Hence, the slope of the given line is 3/7.
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