Point Slope Form: Videos & Practice Problems

Understanding the equation of a line involves mastering both slope intercept form and point slope form. The slope intercept form, $y=mx+b$, uses the slope ($m$) and y-intercept ($b$). Point slope form, $y-y_1=m(x-x_1)$, applies when given a point and slope. Converting between these forms involves distributing the slope and isolating $y$. This skill is essential for graphing lines and solving linear equations effectively.

Point Slope Form of a Line

Understanding how to find the equation of a line is fundamental in algebra, and two common forms used are slope-intercept form and point-slope form. The slope-intercept form is expressed as y = mx + b, where m represents the slope of the line and b is the y-intercept, the point where the line crosses the y-axis. The slope is calculated as the ratio of the rise (change in y) over the run (change in x), which determines the steepness of the line.

While slope-intercept form is straightforward when the slope and y-intercept are known, there are situations where the y-intercept is not given. In such cases, the point-slope form becomes especially useful. This form is written as y - y₁ = m(x - x₁), where (x₁, y₁) is any known point on the line and m is the slope. This equation allows you to write the line’s equation using a specific point and the slope, even if the y-intercept is unknown.

For example, if a line has a slope of 2 and passes through the point (1, 3), the point-slope form would be:

\[y - 3 = 2(x - 1)\]

Converting from point-slope form to slope-intercept form involves distributing the slope across the terms inside the parentheses and then isolating y. For instance, given a slope of m = \frac{1}{2} and a point (-6, -2), the point-slope form is:

\[y - (-2) = \frac{1}{2}(x - (-6))\]

which simplifies to:

\[y + 2 = \frac{1}{2}(x + 6)\]

Distributing the slope yields:

\[y + 2 = \frac{1}{2}x + 3\]

Subtracting 2 from both sides to solve for y gives:

\[y = \frac{1}{2}x + 1\]

This final equation is in slope-intercept form, clearly showing the slope and y-intercept. Mastery of both forms and the ability to convert between them enhances problem-solving flexibility when working with linear equations. Recognizing when to apply point-slope form, especially when given a point and slope but not the y-intercept, is key to efficiently writing the equation of a line.

Write each equation in point-slope form.
Slope: $m=4$; Point: $(1,-2)$

• A. $y-2=4\left(x-1\right)$
• B. $y-4=2\left(x-1\right)$
• C. $y+4=2\left(x-1\right)$
• D. $y+2=4\left(x-1\right)$

Answer: (D)

Write each equation in point-slope form.
Slope: $m=-\frac{3}{2}$; Point: $(0,5)$

• A. $y-5=-\frac32x$
• B. $y=-\frac32\left(x-5\right)$
• C. $y=\frac32\left(x-5\right)$
• D. $y+5=-\frac32x$

Answer: (A)

Write each equation in point-slope form.


Slope: $m=\frac{1}{3}$; Point: $(-2,4)$

• A. $y-4=\frac13\left(x-2\right)$
• B. $y-4=\frac13\left(x+2\right)$
• C. $y-\frac13=4\left(x+2\right)$
• D. $y-\frac13=4\left(x-2\right)$

Answer: (B)

Point Slope Form of a Line Example 1

Graphing a line using a given point and slope involves understanding how the slope represents the ratio of rise over run. For example, if a line passes through the point (3, -1) with a slope of 2, you first locate the point on the coordinate plane. The slope of 2 can be expressed as \(\frac{2}{1}\), meaning from the point (3, -1), you rise 2 units vertically and run 1 unit horizontally to the right to find the next point. Repeating this process creates a series of points that align to form the line. Connecting these points results in the graph of the line with the specified slope passing through the given point.

In another case, if the line passes through the point (-2, 5) with a slope of \(-\frac{1}{4}\), the negative slope indicates a downward rise. Here, the rise is -1 (down 1 unit), and the run is 4 (right 4 units). Starting at (-2, 5), moving down 1 unit and right 4 units locates the next point on the line. Continuing this pattern helps plot additional points, which when connected, form the line with slope \(-\frac{1}{4}\) passing through (-2, 5).

This method of graphing lines emphasizes the practical application of the slope formula, where slope \(m\) is defined as \(m = \frac{\Delta y}{\Delta x}\), representing the change in vertical direction (rise) over the change in horizontal direction (run). By using a known point and the slope, you can accurately plot the line on a coordinate plane, reinforcing the connection between algebraic equations and their graphical representations.

Here’s what students ask on this topic:

What is the point slope form of a line and when should it be used?

Point slope form is an equation used to represent a line when you know a point on the line and the slope. The formula is given by $y-y_1=m(x-x_1)$, where $x_1$ and $y_1$ are the coordinates of the known point, and $m$ is the slope. This form is especially useful when the y-intercept is not known or not given, but you have a point on the line and the slope. It allows you to write the equation of the line directly without needing to find the y-intercept first.

How do you convert an equation from point slope form to slope intercept form?

To convert from point slope form $y-y_1=m(x-x_1)$ to slope intercept form $y=mx+b$, follow these steps: First, distribute the slope $m$ across the parentheses. This gives $y-y_1=mx-m{x}_1$. Next, isolate $y$ by adding $y_1$ to both sides, resulting in $y=mx-m{x}_1+y_1$. This is the slope intercept form, where the constant term $b$ equals $y_1-m{x}_1$.

How do you find the slope of a line given two points?

The slope of a line between two points is found by calculating the change in y-values divided by the change in x-values. If the two points are $(x_1,y_1)$ and $(x_2,y_2)$, the slope $m$ is calculated as $m=\frac{y_2-y_1}{x_2-x_1}$. This ratio represents the rise over run, or how much the line goes up or down for each unit it moves horizontally. Knowing the slope is essential for writing the equation of the line in either point slope or slope intercept form.

What is the difference between slope intercept form and point slope form?

Slope intercept form and point slope form are two ways to write the equation of a line, but they are used in different situations. Slope intercept form is $y=mx+b$, where $m$ is the slope and $b$ is the y-intercept. This form is useful when you know the slope and the y-intercept of the line. Point slope form is $y-y_1=m(x-x_1)$, and it is used when you know the slope and a specific point on the line, but not necessarily the y-intercept. Point slope form is often a starting point to find the slope intercept form by algebraic manipulation.

How do you write the equation of a line using point slope form if given a point and slope?

To write the equation of a line using point slope form when given a point and slope, use the formula $y-y_1=m(x-x_1)$. Here, $x_1$ and $y_1$ are the coordinates of the given point, and $m$ is the slope. Substitute the values of the point and slope into the formula. For example, if the point is (3, 5) and the slope is 2, the equation becomes $y-5=2(x-3)$. This equation represents the line passing through the point with the given slope.