Question

Find the equation of the line parallel to ${\text{3x + 2y = 8}}$ and passing through (0, 1).

Answer 1

Hint: We first find the slope (m) of the given equation. Note that the slope of a line parallel to the given line will also have the same value.
So, we convert the equation of the given line in the form of $y = mx + c$, where m is the slope.
So, the line parallel to the given line will be of the form $y = mx + {c_1}$, where we need to find ${c_1}$.
Now, given that the required line passes through (0, 1)
Therefore, find the value of ${c_1}$ by substituting x=0 and y=1 in its equation. Thus we get the required equation of line.

Complete step-by-step answer:
Consider the given equation ${\text{3x + 2y = 8}}$
$ \Rightarrow 2y = 8 - 3x$
On dividing the equation by 2 we get,
$ \Rightarrow y = - \frac{3}{2}x + 4$
So, this equation is in the form of $y = mx + c$, where m is the slope.
∴ Slope of the given line is $ - \frac{3}{2}$
∴ The slope of the required line is also $ - \frac{3}{2}$ , since it is parallel to the given line, and as parallel lines have the same slope.
Let, the equation of this line be $y = - \frac{3}{2}x + {c_1} \ldots (1)$
Given that, the line passes through (0, 1)
∴ Substituting x=0 and y=1 in equation (1), we get,
$1 = \left( { - \frac{3}{2} \times 0} \right) + {c_1}$
On simplification we get,
 $ \Rightarrow {c_1} = 1$
Therefore,
$\begin{gathered}
  y = - \frac{3}{2}x + {c_1} \\
   \Rightarrow y = - \frac{3}{2}x + 1 \\
\end{gathered} $
On multiplying the equation by 2 we get,
$ \Rightarrow 2y = 2 - 3x$
On rearranging the terms we get,
$ \Rightarrow 3x + 2y = 2$
Hence, the equation of the line parallel to ${\text{3x + 2y = 8}}$ and passing through (0, 1) is $3x + 2y = 2$ .

Note: (Alternative method): The equation of a straight line with slope m and passing through the point (x1, y1) is given by
$y - {y_1} = m(x - {x_1})$
You can directly substitute the values of m, x1 and y1 here to find the equation of the required line.