Find the Equation of a Line

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In order to find the equation of a line, you need two things: a) a point on the line; and b) the slope (sometimes called the gradient) of the line. But how you go about acquiring these two pieces of information, and what you do with them afterwards, can vary depending on the situation. 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).

Steps

General Information

  1. Know what to look for. Before you can find the equation, make sure you have a clear idea of what you're trying to find. Pay attention to these words:
    • Points are identified with ordered pairs such as (-7, -8) or (-2,-6).
    • 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.
  3. Attack the problem using one of the four methods below. Depending on what information you're given, there are different ways to solve it.

Given a Point and a Slope

  1. Calculate the y-intercept of your equation. The y-intercept (or the variable b in our equation) is the point at which the line crosses the y-axis. You can calculate the y-intercept by rearranging the equation to solve for b. Our new equation looks like this: b = y - mx.
    • Plug your slope and coordinates into the above equation.
    • Multiply the slope (m) by the x-coordinate of the point.
    • Subtract that amount FROM the y-coordinate of the point.
    • You've solved for b, or the y-intercept.
  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.
  5. Solve the sample problem. "Given the point (6, -5) and the slope 2/3, what is the equation of the line?"
    • Rearrange your equation. b = y - mx.
    • Plug in and solve.
      • b = -5 - (2/3)6.
      • b = -5 - 4.
      • b = -9
    • Double-check that your y-intercept is really -9.
    • Write down the equation: y = 2/3 x - 9

Given Two Points

  1. Calculate the slope between the two points. Slope is also called "rise over run," and you can think of it as describing how high any line climbs or falls for every unit it travels left or right. The equation for slope is: (Y2 - Y1) / (X2 - X1)
    • Take your two points and plug them into the equation. (Two coordinates means two y values and two x values.) It doesn't matter which coordinates you put first, as long as you stay consistent. Some examples:
      • Points (3, 8) and (7, 12). (Y2 - Y1) / (X2 - X1) = 12 - 8 / 7 - 3 = 4/4, or 1.
      • Points (5, 5) and (9, 2). (Y2 - Y1) / (X2 - X1) = 2 - 5 / 9 - 5 = -3/4.
  2. Choose one set of coordinates for the rest of the problem. Cross out the other set of coordinates or cover it so that you don't accidentally use it.
  3. Calculate the y-intercept of your equation. Again, rearrange the y = mx + b formula to get b = y - mx. It's still the same equation; you've just changed it around.
    • Plug your slope and coordinates into the above equation.
    • Multiply the slope (m) by the x-coordinate of the point.
    • Subtract that amount FROM the y-coordinate of the point.
    • You've solved for b, or the y-intercept.
  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.
  7. Solve the sample problem. "Given the points (6, -5) and (8, -12), what is the equation of the line?"
    • Solve for slope. Slope = (Y2 - Y1) / (X2 - X1)
      • -12 - (-5) / 8 - 6 = -7 / 2
      • 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.)
    • Rearrange your equation. b = y - mx.
    • Plug in and solve.
      • b = -12 - (-7/2)8.
      • b = -12 - (-28).
      • b = -12 + 28.
      • b = 16
      • Note: Since we used the 8 for our coordinates, we must also use the -12. If you use the 6 for your coordinates, then you must also use the -5.
    • Double-check that your y-intercept is really 16
    • Write down the equation: y = -7/2 x + 16

Given a Point and a Parallel Line

  1. Identify the slope of the parallel line. Remember, the slope is the coefficient of x when y does not have a coefficient.
    • 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 = 8, the slope is 3/4.
      • To get the slope on an equation like this, just rearrange it so that y is alone:
      • 4y = 3x + 8
      • Divide both sides by "4": y = 3/4x + 2
  2. Calculate the y-intercept using the slope from the first step and the equation b = y - mx.
    • Plug your slope and coordinates into the above equation.
    • Multiply the slope (m) by the x-coordinate of the point.
    • Subtract that amount FROM the y-coordinate of the point.
    • You've solved for b, or the y-intercept.
  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.
  6. Solve the sample problem. "Given the point (4, 3) and the parallel line 5x - 2y = 1, what is the equation of the line?"
    • Solve for slope. The slope of our new line is going to be the same as the slope of the old line. Figure out the slope of the old line:
      • -2y = -5x + 1
      • Subtract "-2" from both sides: y = 5/2x - 1/2
      • Slope is 5/2.
    • Rearrange your equation. b = y - mx.
    • Plug in and solve.
      • b = 3 - (5/2)4.
      • b = 3 - (10).
      • b = -7.
    • Double-check that your y-intercept is really -7.
    • Write down the equation: y = 5/2 x - 7

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 (or 3/1 — same thing) becomes -1/3
    • -1/2 becomes 2
  3. Calculate the y-intercept using the slope from step 2 and the equation b = y - mx
    • Plug your slope and coordinates into the above equation.
    • Multiply the slope (m) by the x-coordinate of the point.
    • Subtract that amount FROM the y-coordinate of the point.
    • You've solved for b, or the y-intercept.
  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.
  7. Solve the sample problem. "Given (8, -1) and the perpendicular line 4x + 2y = 9, what is the equation of the line?"
    • Solve for slope. The slope of our new line is going to be the negative inverse of the slope of the old line. Figure out the slope of the old line:
      • 2y = -4x + 9
      • Subtract "2" from both sides: y = -4/2x + 9/2
      • Slope is -4/2 or -2.
    • The negative reciprocal of -2 is 1/2.
    • Rearrange your equation. b = y - mx.
    • Plug in and solve.
      • b = -1 - (1/2)8.
      • b = -1 - (4).
      • b = -5.
    • Double-check that your y-intercept is really -5.
    • Write down the equation: y = 1/2 x - 5

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