Parallel lines never meet, so they must be equally steep in the same direction — parallel lines have equal gradients. That single fact is the whole topic; everything else is just applying $y = mx+c$ with a gradient you already know.

1. Identify the gradient of the given line (rearrange to $y=mx+c$ if needed) — your new line has the same $m$.
2. Substitute the known point $(x_1, y_1)$ and this gradient into $y - y_1 = m(x-x_1)$, or substitute directly into $y = mx+c$ and solve for $c$.

Rearrange both equations into $y = mx + c$ form and compare their gradients directly — two lines are parallel if and only if their $m$-values are identical (their $c$-values can be anything). Two lines that look very different when written as, say, $2x + 4y = 8$ and $3x + 6y = 5$ can turn out to be parallel once both are simplified, since both rearrange to a gradient of $-\tfrac12$.

Yes — if both the gradient and the $y$-intercept match, the two equations describe the identical line, not two separate parallel lines. Examiners sometimes test this by giving an equation that simplifies to be identical to another; always check both $m$ and $c$ before concluding two lines are genuinely distinct and parallel.

Parallel-line reasoning is often the final step of a bigger problem rather than the whole question — for example, showing that one side of a quadrilateral is parallel to another to help classify the shape as a trapezium or parallelogram, or confirming that a constructed line matches a given direction. In these cases, calculate both gradients using the two-point formula from E3.3 first, since a "parallel or not" conclusion is only ever one short comparison once both gradients are known.

Find the equation of the line parallel to $y = 4x - 1$ that passes through $(1, -3)$.

1. The given line has gradient $m = 4$. A parallel line has the same gradient, so the new line is $y = 4x + c$ for some $c$.
2. Substitute the point $(1, -3)$ to find $c$:

1. The equation of the parallel line is:

Determine whether the lines $2x + 4y = 8$ and $3x + 6y = 5$ are parallel.

1. Rearrange the first: $4y = 8 - 2x \implies y = 2 - \tfrac12 x$, so $m_1 = -\tfrac12$.
2. Rearrange the second: $6y = 5 - 3x \implies y = \tfrac56 - \tfrac12 x$, so $m_2 = -\tfrac12$.
3. Since $m_1 = m_2 = -\tfrac12$ but the $y$-intercepts ($2$ and $\tfrac56$) are different, the lines are parallel and distinct (not the same line).

Both checks — comparing gradients, then comparing intercepts — are necessary together: matching gradients alone only proves the lines are either parallel or identical, and the intercept comparison is what tells them apart.