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SAT Parallel and Perpendicular Line Slopes

The SAT asks for the equation of a line parallel or perpendicular to a given line, often through a stated point. Two facts carry every one of these questions: parallel lines have equal slopes, and perpendicular lines have slopes that multiply to negative 1. The rest is substitution.

The rule

Parallel lines have equal slopes, and perpendicular slopes multiply to negative 1.

Parallel first. If a line has slope 3, every line parallel to it also has slope 3, and only the y-intercept differs. So a line parallel to y equals 3x minus 8 passing through the point with coordinates 2 and 5 must be y equals 3x plus b. Substituting gives 5 equals 6 plus b, so b equals negative 1 and the line is y equals 3x minus 1.

Perpendicular slopes are negative reciprocals, which means you flip the fraction and change the sign. The perpendicular partner of 3, which is three over one, is negative one third. The partner of negative two fifths is positive five halves. A quick check is to multiply them: 3 times negative one third is negative 1, which is the defining relationship.

Most of the difficulty comes from lines written in standard form. Given 4x plus 2y equals 9, do not guess the slope from the coefficients. Solve for y: 2y equals negative 4x plus 9, so y equals negative 2x plus 4.5 and the slope is negative 2. A parallel line has slope negative 2, and a perpendicular line has slope positive one half.

Horizontal and vertical lines are the special case worth knowing in advance. A horizontal line has equation y equals a constant and slope 0. A vertical line has equation x equals a constant and no defined slope at all. They are perpendicular to each other even though the negative reciprocal rule cannot be applied, since you cannot flip a slope of zero.

Some questions never ask for an equation. They give two lines and ask whether they are parallel, perpendicular, or neither, or they give one slope and ask for the other. Compute both slopes, then compare: equal means parallel, a product of negative 1 means perpendicular, and anything else means the lines cross at some other angle.

The trap

Taking the negative of the slope without flipping the fraction, so the perpendicular partner of two thirds gets called negative two thirds instead of negative three halves.

Try it

Try one
Real question

Which is NOT on the line y = 2x - 3?

A laptop battery's use satisfies 9v + 4b = 468, where v is the number of minutes of video played and b is the number of minutes of background work. If 32 minutes of video are played on one charge, how many minutes of use in all does that charge give?

  • A) 32
  • B) 45
  • C) 77 ✓
  • D) 117

Why: The video uses 9(32) = 288, leaving 180 for background work, which is 180/4 = 45 minutes. The charge gives 32 + 45 = 77 minutes of use. Choice B counts only the background minutes. Answering 32 means you gave the video minutes only. Answering 45 means you gave the background minutes only. Answering 117 means you divided 468 by 4 without subtracting the video's share first.

Practice more of these free

Questions students ask

How do I find the slope of a line in standard form?

Solve the equation for y. From ax plus by equals c you get y equals negative a over b times x plus c over b, so the slope is negative a divided by b. Doing the algebra once is safer than memorizing the formula, especially when b is negative.

Are vertical and horizontal lines perpendicular?

Yes, they meet at a right angle. The negative reciprocal test does not apply because a vertical line has undefined slope and a horizontal line has slope zero, and you cannot take the reciprocal of zero. Treat this pair as a known exception.

Guides that cover this

More on linear equations in two variables