How to Find a Parallel Line to y -3x 4 Passing Through the Point (3, 4)

Understanding how to find a line parallel to another and passing through a specific point is a valuable skill in mathematics and has numerous real-world applications, including in graphic design, computer graphics, and engineering. This guide will walk you through the process using the point-slope formula. Let's explore the steps to determine the equation of a line parallel to y -3x 4 and passing through the point (3, 4).

Step-by-Step Guide

When dealing with parallel lines, it's important to remember that parallel lines have the same slope. In our case, the given line is y -3x 4, which is in slope-intercept form (y mx b), where m is the slope and b is the y-intercept.

Identifying the Slope

The slope (m) of the given line y -3x 4 is -3. Any line parallel to this line will have the same slope.

Using the Point Slope Formula

The point-slope formula is used to find the equation of a line when we know one point through which the line passes and the slope of the line. The point-slope formula is:

y - y? m(x - x?)

Where:

y is the dependent variable. x is the independent variable. m is the slope. (x?, y?) is a point on the line.

Plugging in the Known Values

We know from the given problem that the slope (m) is -3. We also know that the line passes through the point (3, 4). We can plug these values into the point-slope formula:

y - 4 -3(x - 3)

Simplifying the Equation

Now, let's simplify the equation step-by-step:

Expand the right side of the equation:

y - 4 -3x 9

Add 4 to both sides to solve for y:

y -3x 9 4

Simplify the right side:

y -3x 13

Therefore, the equation of the line parallel to y -3x 4 and passing through the point (3, 4) is y -3x 13.

Conclusion

To summarize, the process of finding a parallel line involves using the point-slope formula, knowing the slope from the given line, and plugging in the point through which the new line passes. Understanding this process not only helps in solving mathematical problems but also enhances problem-solving skills in broader fields.

Related Keywords

parallel lines, point-slope formula, slope-intercept form

Further Resources

To deepen your understanding of parallel lines and related concepts, you can explore resources such as:

Online tutorials on algebra and geometry. Math textbooks focusing on linear equations and graphing. Interactive math tools and software for visualizing lines and equations.