How to Determine the Equation of a Line Given its Slope and a Point

When you are tasked with finding the equation of a line that passes through a given point and has a certain slope, you can use the point-slope form to achieve your goal. In this article, we will walk through the process step-by-step, using a specific example to illustrate the method.

Point-Slope Form of a Linear Equation

The point-slope form of a linear equation is a useful tool for this purpose. It is given by the formula:

y - y1 m(x - x1)

Here, m represents the slope of the line and (x1, y1) is a point through which the line passes. This formula allows you to derive the equation of a line when you know the slope and a point.

Applying the Formula to a Specific Example

Let's consider a specific example where the slope (m) is 7 and the line passes through the point (-8, 3).

Step 1: Plug the slope and the point into the point-slope formula:

y - 3 7(x - (-8))

Step 2: Simplify the expression:

y - 3 7(x 8)

Step 3: Distribute the 7 on the right-hand side:

y - 3 7x 56

Step 4: Add 3 to both sides to solve for y:

y 7x 59

Equation of the Line in Different Forms

Let's explore the line equation in its various forms.

Slope-Intercept Form

The slope-intercept form of a line is:

y mx b

From our previous simplification, we have:

y 7x 59

In this form, m (the slope) is 7, and b (the y-intercept) is 59.

Standard Form

The standard form of a linear equation is:

ax by c

From the previous steps, we can rearrange the equation:

7x - y 59 0

This is now in standard form where a 7, b -1, and c 59.

Using the Correct Formula and Solving for b

Let's go through the process again to ensure we get the correct equation of the line:

Given: m 7 and point (-8, 3)

Using the point-slope formula:

y - 3 7(x 8)

Distribute the 7:

y - 3 7x 56

Add 3 to both sides:

y 7x 59

Conclusion

Therefore, the equation of the line with a slope of 7 and passing through the point (-8, 3) is:

y 7x 59 or in standard form: 7x - y 59 0

This method can be applied to other examples with similar information, such as a slope and a point. By understanding the point-slope form, you can derive the equation of a line and move it into other forms as needed.

Further Resources

For a more comprehensive understanding and practical examples, you can use software like WolframAlpha or Geogebra to visualize these lines and points. These tools provide a valuable resource for visual learners and those who want to explore the visual representation of equations.