Understanding the Linear Equation of a Line with a Slope of 1/2 and Y-Intercept of 10

Understanding the Linear Equation of a Line with a Slope of 1/2 and Y-Intercept of 10

In linear algebra, the equation of a line is often expressed in the slope-intercept form, which is y mx b. Here, m represents the slope of the line, indicating how steep the line is, and b is the y-intercept, the point at which the line crosses the y-axis.

Deriving the Equation

We are given a line that has a slope of 1/2 and a y-intercept of 10. Using the slope-intercept form, we can substitute these values into the equation:

y mx b

Substituting m 1/2 and b 10:

y (1/2)x 10

This is the equation of the line that meets the given criteria.

Alternative Forms of the Equation

The equation y (1/2)x 10 can be manipulated to express it in different forms:

Simplifying the Equation

Multiplying both sides of the equation by 2 to clear the fraction:

2y x 20

By rearranging terms, we can write it in the general form of a linear equation:

x - 2y - 20 0

Or, in standard form:

x - 2y -20

These alternative forms are equally valid and can be helpful in different contexts.

Variants and Contexts

Let's explore a few more ways to represent the same equation:

Standard Variations

Given the equation in its simplest form:

y (1/2)x 10

We can also write it in a slightly different format:

y x/2 10

Multiplying both sides by 2 again:

2y x 20

General Context

In a more general sense, the linear equation y (1/2)x 10 can be further explored with the general form of a line, which is ax by c 0:

By rearranging y (1/2)x 10, we get:

x - 2y - 20 0

Here, a 1, b -2, and c -20.

Conclusion

To summarize, the linear equation of a line with a slope of 1/2 and a y-intercept of 10 is y (1/2)x 10. This can also be expressed as 2y x 20, x - 2y - 20 0, or x - 2y -20, all of which are equivalent forms. Understanding these different representations enhances our ability to work with and solve linear equations in various contexts.