How to Use Point-Slope Form to Find a Line’s Equation
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 Point Slope Line Equation Explained for Students
1. What is the point-slope form of a line equation?
The point-slope form is a way to write the equation of a straight line using its slope and the coordinates of a single point on the line. The standard formula is y - y₁ = m(x - x₁). In this formula:
- m represents the slope of the line.
- (x₁, y₁) are the coordinates of the known point on the line.
- (x, y) represents the coordinates of any other point on the line.
2. How do you find the equation of a line with a given point and slope?
To find the equation of a line using the point-slope form, you can follow these simple steps:
- Identify your known values: Determine the slope (m) and the coordinates of the given point (x₁, y₁).
- Substitute into the formula: Plug these values directly into the point-slope equation: y - y₁ = m(x - x₁).
- Simplify the equation: Perform the necessary algebraic operations to simplify the equation, often into the slope-intercept form (y = mx + b) for easier interpretation.
3. When is it better to use point-slope form instead of slope-intercept form (y = mx + c)?
It is better to use the point-slope form when you are given the slope of a line and any point on that line which is not the y-intercept. The slope-intercept form, y = mx + c, is most efficient only when you are explicitly given the slope and the y-intercept (the point where the line crosses the y-axis, i.e., (0, c)). The point-slope form is more versatile as it works with any point on the line, not just a specific one.
4. How does the point-slope formula relate to the basic definition of slope?
The point-slope formula is a direct rearrangement of the basic slope definition. The formula for the slope (m) between two points (x₁, y₁) and (x₂, y₂) is m = (y₂ - y₁) / (x₂ - x₁). If we consider (x₁, y₁) as our known fixed point and let (x, y) be any other variable point on the line (in place of (x₂, y₂)), the slope formula becomes m = (y - y₁) / (x - x₁). By multiplying both sides by (x - x₁), you arrive directly at the point-slope form: y - y₁ = m(x - x₁). This shows that the point-slope form is just the slope formula in a different algebraic arrangement.
5. What is the main difference between the point-slope form and the two-point form?
The main difference lies in the initial information required to use them:
- The point-slope form requires one point (x₁, y₁) and the slope (m).
- The two-point form requires two different points (x₁, y₁) and (x₂, y₂).
6. Can the point-slope form be used to describe horizontal and vertical lines?
The point-slope form handles horizontal and vertical lines differently:
- Horizontal Lines: Yes. A horizontal line has a slope (m) of 0. Substituting m = 0 into the point-slope formula gives y - y₁ = 0(x - x₁), which simplifies to y = y₁. This correctly represents the equation of a horizontal line.
- Vertical Lines: No. A vertical line has an undefined slope. Since the value of 'm' is undefined, you cannot substitute it into the point-slope formula. The equation of a vertical line is always given in the form x = x₁, where x₁ is the constant x-coordinate for all points on that line.
