Parallel Lines

2 MIN READ · PRACTISE AT YOUR OWN PACE · FREE

Idea

Parallel lines never meet. They never meet because they rise at exactly the same rate.

Same steepness, different position. That is the whole topic: equal slopes, different intercepts.

Rule

Two distinct lines are parallel exactly when their slopes are equal:

m_1 = m_2 \quad \text{and} \quad n_1 \ne n_2

In standard form the same condition reads as matching coefficient ratios:

\frac{a_1}{a_2} = \frac{b_1}{b_2} \ne \frac{c_1}{c_2}

The second requirement matters. Drop it and you have described coincident lines, not parallel ones.

How it is used

The usual task is this: build a line parallel to a given line, passing through a given point.

Take the slope from the line you were given - parallel means you inherit it unchanged.
Use the point you were given with the point-slope form.
Confirm it is a different line, by checking the point is not already on the original.

There is a shortcut for standard form.

Any line parallel to $ax + by + c = 0$ looks like $ax + by + k = 0$ . Keep both coefficients and change only the constant. Put your point in to find $k$ , and you are done in one step.

Two vertical lines are parallel too, even though neither has a slope. The ratio test still works here, which is a good reason to use it.

Worked example

Find the line through $(2, 1)$ parallel to $3x - y + 2 = 0$ . First check the point is not already on that line: $3(2) - 1 + 2 = 7$ , which is not zero, so a genuine parallel exists. Parallel means both coefficients survive, so the answer has the form $3x - y + k = 0$ . Substitute the point to pin down $k$ : $3(2) - 1 + k = 0$ , so $5 + k = 0$ and $k = -5$ .

3x - y - 5 = 0

Explore

Set the blue line first. Then drag C and D until the verdict says Parallel and the two slopes match. The meeting point disappears as soon as they agree, and comes back if they differ by one step.

Two lines, one plane

Try this. Use A(0,0), B(1,1), C(0,1), D(1,2) for parallel lines. Change D to (1,0) for perpendicular lines.

Point A · x-2Point A · y3Point B · x3Point B · y1Point C · x-3Point C · y-2Point D · x3Point D · y4

Blue slope: -0.4 · Green slope: 1 · Intersecting at (0.857, 1.857).

PAUSE & THINKA quick check, not a grade

Try it yourself.

Choose an answer and explain your reasoning to yourself. Use a hint if you get stuck.

Which line is parallel to

y=-3x+2

$y=x/3+2$

$y=3x-4$

$y=-3x-4$

Hint 1 · Find a starting point

Parallel nonvertical lines have equal slopes.

Hint 2 · Take the next step

Look only at the coefficient of x to compare directions.

Show the reasoning

Answer: $y=-3x-4$

The slope must remain −3. Changing the intercept translates the line without changing its direction.

Before moving on: what would make one of the other answers wrong? Saying why is part of understanding.

MAKE IT YOURS

Pause before the next idea.

Can you explain how the visualization connects to this lesson’s goal? If a step still feels uncertain, put this lesson on your review list and try the check again another day.

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