Converting the Equation into Symmetric Form

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To convert the given equation \(x – \sqrt{3}y = 1\) into its symmetric form, follow these steps:

Rewrite the given equation: \(x – \sqrt{3}y = 1\)

Identify the intercepts:

• • Set \(y = 0\) to find the x-intercept (\(a\)):

\(x = 1\)

So, \(a = 1\).

• • Set \(x = 0\) to find the y-intercept (\(b\)):

\(-\sqrt{3}y = 1 \implies y = -\frac{1}{\sqrt{3}}\)

So, \(b = -\frac{1}{\sqrt{3}}\).

Express the equation in symmetric form:

\(\frac{x}{a} + \frac{y}{b} = 1\)

Substituting the values of \(a\) and \(b\):

\(\frac{x}{1} + \frac{y}{-\frac{1}{\sqrt{3}}} = 1\)

Simplify the y-term: \(\frac{x}{1} – \sqrt{3}y = 1\)

Notice that the simplified symmetric form directly comes from rearranging the given equation with clear intercepts.

Thus, the symmetric form of the equation \(x – \sqrt{3}y = 1\) is:

\(\frac{x – 1}{1} = \frac{y}{\frac{1}{\sqrt{3}}}\)

Or equivalently:

\(\frac{x}{1} – \frac{y}{\frac{1}{\sqrt{3}}} = 1\)

Plot of the line \(x – \sqrt{3}y = 1\)

Plot of the line \(x – \sqrt{3}y = 1\), with labels for the x-intercept \((1, 0)\) and the y-intercept \((0, -\frac{1}{\sqrt{3}})\). The x-intercept is marked in red, and the y-intercept is marked in green.