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.