Condition for Straight Lines in Plane to be Parallel/Examples/Arbitrary Example 1

Examples of Use of Condition for Straight Lines in Plane to be Parallel

Let $\LL_1$ be the straight line whose equation in general form is given as:

$3 x - 4 y = 7$

Let $\LL_2$ be the straight line parallel to $\LL_1$ which passes through the point $\tuple {1, 2}$.

The equation for $\LL_2$ is:

$3 x - 4 y = -5$

Proof

From Condition for Straight Lines in Plane to be Parallel, $\LL_2$ has an equation of the form:

$(1): \quad 3 x - 4 y = C$

We have that $\tuple {1, 2}$ is on $\LL_2$.

Hence substituting $x = 1$ and $y = 2$ into $(1)$:

\(\ds C\)

\(=\)

\(\ds 3 \paren 1 - 4 \paren 2\)

\(\ds \)

\(=\)

\(\ds -5\)

Hence the result.

$\blacksquare$

Sources

• 1953: L. Harwood Clarke: A Note Book in Pure Mathematics ... (previous) ... (next): $\text {III}$. Analytical Geometry: The Straight Line: The angle between two lines: Example $\text{(i)}$