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[edit] Steps

1. Know your vocabulary.
   • Points are identified with ordered pairs such as (3, -4) or (-6, 1) or (0,5).
   • The first number in an ordered pair is the x-coordinate. It controls the point's horizontal position (how much to the right or left of the origin).
   • The second number in an ordered pair is the y-coordinate. It controls the point's vertical position (how much up or down from the origin).
   • The slope between two points is defined as "rise over run" --- in other words, the description of how far you must travel up (or down) and to the right (or left) in order to move from one point to the other.
   • Two lines are parallel if they do not intersect (cross over each other).
   • Two lines are perpendicular if they intersect to form a right angle (90 degrees).
2. Identify the type of problem.
   • You are given a point and a slope.
   • You are given two points but no slope.
   • You are given a point and another line that is parallel to yours.
   • You are given a point and another line that is perpendicular to yours.

For simplicity's sake, this article will focus on the slope-intercept equation y = mx + b instead of the point-slope form (y - y₁) = m(x - x₁).

[edit] Given a point and a slope

1. Calculate the y-intercept of your equation using the modified version of the slope-intercept equation, b = y - mx
   • multiply the slope by the x-coordinate of the point
   • subtract that amount FROM the y-coordinate of the point
2. Write out the formula: y = ____ x + ____ , including the blanks.
3. Fill the first blank, in front of the x, with the slope.
4. Fill the second blank with the y-intercept that you calculated earlier.

Example: Given the point (6, -5) and the slope 2/3

• 2/3 x 6 = 4
• -5 subtract 4 = -9
• y-intercept is -9
• equation is: y = 2/3 x - 9

[edit] Given two points

1. Calculate the slope between the two points. For more information on how to do this, consult the article How to Understand Slope (in Algebra)
2. Choose one point (you may want to cross out the other) for the rest of the problem.
3. Calculate the y-intercept of your equation using the formula b = y - mx
   • multiply the slope with the x-coordinate of the point
   • subtract that amount FROM the y-coordinate of the point
4. Write out the formula: y = ____ x + ____ , including the blanks.
5. Fill the first blank, in front of the x, with the slope.
6. Fill the second blank with the y-intercept.

Example: Given the points (6, -5) and (8, -12)

• slope is -7/2
  • from the first point to the second, we went down 7 and right 2, so the slope is -7 over 2
• -7/2 x 8 = -28
• -12 - -28 = 16
  • NOTE: Since we used the 8 on step 2, we must use the -12 on step 3. If you use the 6 on step 2, then you must use the -5 on step 3.
• y-intercept is 16
• equation is: y = -7/2 x + 16

[edit] Given a point and a parallel line

1. Identify the slope of the parallel line.
   • In an equation like y = 3/4 x + 7, the slope is 3/4.
   • In an equation like y = 3x - 2, the slope is 3.
   • In an equation like y = 3x, the slope is still 3.
   • In an equation like y = 7, the slope is zero (because there are zero x's in the problem).
   • In an equation like y = x - 7, the slope is 1.
   • In an equation like -3x + 4y = 7, the slope is 3/4.
     • When x and y are on the same side, the shortcut is: slope = coeff of x, over coeff of y, with a sign change.
   • In an equation like 2x - 5y = 10, the slope is 2/5.
   • In an equation like 6x + 7y = 1, the slope is -6/7.
   • In an equation like -3x - 4y = 8, the slope is -3/4.
   • In an equation like 4x - y = 9, the slope is 4.
     • NOTE: if the signs of the coefficients are the same, the slope is positive. If different, then the slope is negative.
2. Calculate the y-intercept using the slope from step 1 and the equation b = y - mx
   • multiply the slope from step 1 with the x-coordinate of the point
   • subtract that amount FROM the y-coordinate of the point
3. Write out the formula: y = ____ x + ____ , including the blanks.
4. Fill the first blank, in front of the x, with the slope you identified on step 1. The deal with parallel lines is that they have the same slope, so what you started with is also what you end with.
5. Fill the second blank with the y-intercept.

Example: Given the point (4, 3) and the parallel line 5x - 2y = 1

• slope is 5/2
• 5/2 x 4 = 10
• 3 - 10 = -7
• y-intercept is -7
• equation is: y = 5/2 x - 7

[edit] Given a point and a perpendicular line

1. Identify the slope of the given line. Consult the examples above for more information.
2. Find the negative reciprocal of that slope. In other words, flip it over and change the sign. The deal with perpendicular lines is that they have negative reciprocal slopes, so you have to make changes to the slope before you can use it.
   • 2/3 becomes -3/2
   • -6/5 becomes 5/6
   • 3 becomes -1/3
   • -1/2 becomes 2
3. Calculate the y-intercept using the slope from step 2 and the equation b = y - mx
   • multiply the slope from step 2 with the x-coordinate of the point
   • subtract that amount FROM the y-coordinate of the point
4. Write out the formula: y = ____ x + ____ , including the blanks.
5. Fill the first blank, in front of the x, with the slope you calculated in step 2.
6. Fill the second blank with the y-intercept.

Example: Given (8, -1) and the perpendicular line 4x + 2y = 9

• slope of the original line is -4/2 or just -2
• neg. reciprocal is 1/2
• 1/2 x 8 = 4
• -1 - 4 = -5
• y-intercept is -5
• equation is: y = 1/4 x - 5

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