Intercept Form of a Line: Exploring Slopes and Equations

In this article, we'll explore the properties of a line defined by its intercepts on the x-axis and y-axis. Specifically, we will derive and compare the equations of lines with given intercept conditions and explore their slopes. This understanding is crucial for anyone working with linear equations and graphical representations in mathematics or related fields.

Understanding Intercepts

Consider a line L that intercepts the Y-axis at a Y-intercept a and intercepts the X-axis at an X-intercept 2a. This means the line passes through the points (0, a) and (2a, 0).

Deriving the Equation of Line L

Given the intercepts, the equation of line L can be written in its intercept form:

[frac{x}{2a} frac{y}{a} 1]

By multiplying through by 2a, we obtain a more familiar linear form:

[x 2y 2a]

Using the Points (3, 4) to Derive Specific Equations

Let's use the point (3, 4) to determine the specific value of a and the equation of the line. Substituting (3, 4) into the equation:

[3 2(4) 2a]

Solving for a gives:

[3 8 2a 11 2a a 5.5]

Thus, the equation of the line is:

[x 2y 11]

Alternative Approaches and Equivalent Forms

Alternatively, we can rearrange the equation to match the form:

[2y -frac{x}{2} 11 y -frac{x}{2} frac{11}{2}]

Another way to present the same equation is to multiply through by 2:

[2y -x 11 x 2y 11]

Comparing Solutions Using Different Methods

We can also solve for a using different methods, such as the slope-intercept form:

[y mx b]

Given that the line passes through (3, 4), we can establish that:

[4 m(3) b]

And knowing that the x-intercept is 2a, we can use the point (2a, 0) to find b:

[0 m(2a) b b -2am]

Substituting b back into the first equation:

[4 3m - 2am 4 m(3 - 2a)]

Solving for m and a

[m frac{4}{3 - 2a}]

To satisfy the condition that the x-intercept is 2a, we equate:

[2a -frac{b}{m} 2a]

This leads to the solution for a as before:

[a 5.5]

The equation of the line is then: