Point Slope Form Formula Derivation and Solved Examples
What is a Line?
In geometry, to describe straight objects with negligible width and depth, the notion of line or straight line was introduced by ancient mathematicians. Lines are an idealization of certain objects that are often represented or referred to with a single letter in terms of two points.
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What is a Point?
A point typically refers to a part called space in a certain set. More precisely, a point is a primitive notion in Euclidean geometry on which the geometry is constructed, meaning that a point can not be described in terms of objects previously defined.
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What is a Slope?
The slope or gradient of a line in mathematics is a number that defines both the direction and the steepness of the line Slope is often referred to by the letter m; there is no straightforward answer to the question of why the letter m is used for slope, but in O'Brien who wrote the straight-line equation as
y = mx + b, its earliest use in English appears.
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How to Find a Point on a Line?
Select x and solve the equation for y, or the equation for y.
Select y and solve for x.
How to Find the Slope of a Line Equation?
The ratio of the "vertical change" to the "horizontal change" between (any) two distinct points on a line is determined by finding the slope. The ratio is often represented as a quotient ("rise over run"), giving the same number on the same line for any two distinct points. A line that decreases has a "rise" that is negative.
Different Forms of the Equation of a Line
Equations of horizontal and vertical lines
Point-slope form equation of a line
Two-point form equation of a line
Slope-intercept form equation of a line
Intercept form
Normal form
What is the Point Slope Equation of a Line?
We can learn how to find the equation of a line given one point and slope that is inclined at a given angle to the positive direction of the x-axis in the anticlockwise sense and passes through a given point by the equation of a line in point-slope form.
Let the MN line form an angle with the positive x-axis direction in the anticlockwise sense and pass through the Q point (x1, y1). We need to find an equation for the MN line. Let any point on the line MN be P(x, y). But Q (x1, y1) is a point on the same side as well.
Therefore, the slope of the MN line = (y-y1) / (x-x1)
Again, the MN line produces an angle with the positive direction of the x-axis; thus, the line slope = tan = m (say).
Therefore, (y-y1) / (x-x1) = m
⇒ y-y1 = m (x-x1)
The equation y-y1 = m (x-x1) above is fulfilled by the coordinates of any point P on the line MN.
Therefore, y-y1 = m (x-x1) represents the AB straight line equation.
Solved Examples
Write the line's point-slope form with the value of slope being 3 that goes through the point (2,5).
Solution: The slope is given as m=3, and there are coordinates of x1 = 2 and y1 = 5 for the point (2,5). To get the final answer, plug the known values into the slope-intercept form now.
y-y1 = m (x-x1)
Y-5 = 3(x-2)
A straight line passes through the point (2, -3) and the positive orientation of the x-axis gives an angle of 135 °. Find the Straight Line Equation.
Solution: An angle of 135 ° with the positive direction of the axis of x renders the appropriate line.
The slope of the appropriate line, therefore,
m= tan 135 ° = tan (90 ° + 45 °) = - cot 45 ° = -1.
Again, the line that is needed passes through the point (2, -3).
We know that a straight line formula passes through a given point (x1, y1) and that the slope of 'm' is y-y1 = m (x-x1).
Therefore, the necessary straight line formula is y - (-3) = -1 (x -2)
y + 3 = -x + 2
x + y + 1 = 0
FAQs on Line Equation in Point Slope Form Explained Clearly
1. What is the point-slope form of a line?
The point-slope form of a line is y − y₁ = m(x − x₁), where m is the slope and (x₁, y₁) is a point on the line.
- m represents the slope (rate of change).
- (x₁, y₁) is a known point on the line.
- This form is useful when you know one point and the slope.
2. How do you write an equation in point-slope form?
To write an equation in point-slope form, substitute the given slope and point into y − y₁ = m(x − x₁).
- Step 1: Identify the slope m.
- Step 2: Identify the point (x₁, y₁).
- Step 3: Substitute into the formula.
3. What is the formula for point-slope form?
The formula for the point-slope form of a linear equation is y − y₁ = m(x − x₁).
- m = slope of the line
- (x₁, y₁) = known point on the line
4. How do you convert point-slope form to slope-intercept form?
To convert point-slope form to slope-intercept form, expand and solve for y.
- Start with y − y₁ = m(x − x₁).
- Distribute m: y − y₁ = mx − mx₁.
- Add y₁ to both sides.
y − 5 = 3x − 6
y = 3x − 1
The slope-intercept form is y = 3x − 1.
5. How do you find the slope for point-slope form?
The slope used in point-slope form is calculated as m = (y₂ − y₁)/(x₂ − x₁).
- Choose two points (x₁, y₁) and (x₂, y₂).
- Subtract the y-values.
- Subtract the x-values.
- Divide the differences.
6. Can you give an example of a line written in point-slope form?
An example of a line in point-slope form is y − 4 = −2(x − 1).
- The slope m = −2.
- The point is (1, 4).
7. What is the difference between point-slope form and slope-intercept form?
The main difference is that point-slope form uses a point and slope, while slope-intercept form shows the slope and y-intercept.
- Point-slope form: y − y₁ = m(x − x₁)
- Slope-intercept form: y = mx + b
- b represents the y-intercept.
8. Why is point-slope form useful?
Point-slope form is useful because it quickly creates a linear equation when you know one point and the slope.
- No need to calculate the y-intercept first.
- Direct substitution into y − y₁ = m(x − x₁).
- Helpful in coordinate geometry and algebra problems.
9. How do you graph a line from point-slope form?
To graph a line from point-slope form, plot the given point and use the slope to find another point.
- Step 1: Identify the point (x₁, y₁).
- Step 2: Plot that point on the coordinate plane.
- Step 3: Use the slope m (rise/run) to find another point.
- Step 4: Draw a straight line through the points.
10. What are common mistakes when using point-slope form?
Common mistakes in point-slope form include sign errors and incorrect substitution.
- Forgetting the formula is y − y₁ = m(x − x₁), not y + y₁.
- Not subtracting inside the parentheses correctly.
- Mixing up x₁ and y₁ values.
- Incorrectly distributing the slope.
