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To find the equation of a line, you need two things : a) a point on the line; and b) its slope (sometimes called its slope). But the way you find this information and what you can do with it later may vary from case to case. For simplicity, this article will focus on equations that take the form of slope and origin y = mx + b instead of the slope form and a point on the line (y – y 1 ) = m(x) – x 1 ) .
Steps
General information
Know what you’re looking for. Before you start looking for equations, make sure you clearly understand what you’re trying to find. Pay attention to the following sentences:
- Scores are determined with ordered pairs such as (-7, -8) or (-2,-6).
- The first number in the ordered pair is the coordinate . It controls the horizontal position of the point (how much left or right it is relative to the origin).
- The second number in the ordered pair is the coordinate . It controls the vertical position of the point (how much above or below the origin).
- The slope between two points is defined as “straight over horizontal” — in other words, it represents how far you have to go up (or down) and right (or left) to get from point to point of the line.
- Two lines are parallel if they do not intersect (intersect).
- Two lines are perpendicular to each other if they intersect and form a right angle (90 degrees).
Determine the problem type.
- Know the slope and a point.
- Know two points on the line but give no slope.
- Know a point on a line and another line parallel to that line.
- Know a point lies on a line and another line is perpendicular to that line.
Solve the problem using one of the four methods shown below. Depending on the information given, we have different solutions.
Know the slope and a point on the line
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Calculate the intercept in your equation. The origin (or b variable in the equation) is the intersection of the line and the vertical axis. You can calculate the origin by rearranging the equation, and finding b . Our new equation looks like this: b = y – mx.
- Enter the slope and coordinates in the above equation.
- Multiply the slope ( m ) by the coordinate of the given point.
- MINUS the product of the coordinates of the point.
- You have found b , or the origin of the equation.
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Write the formula: y = ____ x + ____ , with spaces.
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Fill in the first space, preceded by x, with the slope.
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Fill in the second space with the slope you just calculated.
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Solve the example problem. “Find the equation of the line that passes through the point (6, -5) and has a slope of 2/3.”
- Rearrange the equation. b = y – mx.
- Substitute values and solutions.
- b = -5 – (2/3)6.
- b = -5 – 4.
- b = -9
- Double check if your origin is really -9.
- Write the equation: y = 2/3 x – 9
Knowing two points lying on the line
Calculate the slope between those two points. The slope is also known as “straight over horizontal” and you can think of it as a description of how much the line has gone up or down by moving left or right by one unit. The equation for the slope is: (Y 2 – Y 1 ) / (X 2 – X 1 )
- Take two known points and substitute them into the equation (The two coordinates here are two y values and two x values). It doesn’t matter which coordinates you put in first, as long as you’re consistent in your substitutions. Here are a few examples:
- Scores (3, 8) and (7, 12) . (Y 2 – Y 1 ) / (X 2 – X 1 ) = 12 – 8 / 7 – 3 = 4/4, or 1.
- Score (5, 5) and (9, 2) . (Y 2 – Y 1 ) / (X 2 – X 1 ) = 2 – 5 / 9 – 5 = -3/4.
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Choose a coordinate pair for the rest of the problem. Cross out the remaining coordinate pairs or cover them so you don’t accidentally use them.
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Calculate the origin of the equation. Again, rearrange the formula y = mx + b to get b = y – mx. It’s still the same equation, you just changed it up a bit.
- Generation of angle numbers and coordinates into the above equation.
- Multiply the slope ( m ) by the coordinate of the point.
- Take the coordinates of the point MINUS the above product.
- You just found b , or the intercept.
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Write the formula: y = ____ x + ____ ‘ , including spaces.
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Fill in the slope in the first space, preceded by x.
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Fill in the zero offset in the second space.
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Solve the example problem. “Given two points (6, -5) and (8, -12). Find the equation of the line that passes through these two points.”
- Find the slope. Coefficient of slope = (Y 2 – Y 1 ) / (X 2 – X 1 )
- -12 – (-5) / 8 – 6 = -7 / 2
- The slope is -7/2 (From the first point to the second, we go down 7 and to the right 2, so the slope is – 7 out of 2).
- Rearrange your equation. b = y – mx.
- Numbers and solutions.
- b = -12 – (-7/2)8.
- b = -12 – (-28).
- b = -12 + 28.
- b = 16
- Note : When substituting coordinates, since you used 8, you must also use -12. If you use 6, you will have to use -5.
- Double check to make sure your slope is actually 16.
- Write the equation: y = -7/2 x + 16
Knowing a point and a parallel line
Determine the slope of the parallel line. Remember that the slope is the coefficient of x and y has no coefficients.
- In the equation y = 3/4 x + 7, the slope is 3/4.
- In the equation y = 3x – 2, the slope is 3.
- In the equation y = 3x, the slope is still 3.
- In the equation y = 7, the slope is zero (because the problem doesn’t have x).
- In the equation y = x – 7, the slope is 1.
- In the equation -3x + 4y = 8, the slope is 3/4.
- To find the slope of the above equation, simply rearrange the equation so that y stands alone:
- 4y = 3x + 8
- Divide both sides by “4”: y = 3/4x + 2
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Calculate the origin using the slope found in the first step and the equation b = y – mx.
- Generation of angle numbers and coordinates into the above equation.
- Multiply the slope ( m ) by the coordinate of the point.
- Take the coordinates of the point MINUS the above product.
- You’ve just found b , the slope of the origin.
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Write the formula: y = ____ x + ____ , including spaces.
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Fill in the slope just found in step 1 in the first space, before x. The problem with parallel lines is that they have the same slope, so your starting point is also your ending point.
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Fill in the zero offset in the second space.
Solve the same problem. “Find the equation of the line that passes through the point (4, 3) and is parallel to the line 5x – 2y = 1”.
- Find the slope. The slope of our new line is also the slope of the old line. Find the slope of the old line:
- -2y = -5x + 1
- Divide both sides by “-2”: y = 5/2x – 1/2
- The slope is 5/2 .
- Rearrange the equation. b = y – mx.
- Numbers and solutions.
- b = 3 – (5/2)4.
- b = 3 – (10).
- b = -7.
- Double check to make sure that -7 is exactly the origin.
- Write the equation: y = 5/2 x – 7
Knowing a point and a perpendicular line
Determine the slope of the given line. Review the previous examples for more information.
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Find the opposite inverse of that slope. In other words, reverse the number and change the sign. The problem with two perpendicular lines is that they have opposite inverse slopes. Therefore, you must transform the slope before using it.
- 2/3 becomes -3/2
- -6/5 becomes 5/6
- 3 (or 3/1 — the same) becomes -1/3
- -1/2 becomes 2
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Calculate the slope using the slope in step 2 and the equation b = y – mx
- Generation of angle numbers and coordinates into the above equation.
- Multiply the slope ( m ) by the coordinate of the point.
- Take the coordinates of the point MINUS this product.
- You have found b , the slope of the origin.
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Write the formula: y = ____ x + ____ ‘ , including spaces.
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Fill in the slope calculated in step 2 in the first space, preceded by x.
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Fill in the zero offset in the second space.
Solve the same problem. “Give a point (8, -1) and a line 4x + 2y = 9. Find the equation of the line that passes through that point and is perpendicular to the given line”.
- Find the slope. The slope of the new line is the opposite inverse of the slope of the given line. We find the slope of the given line as follows:
- 2y = -4x + 9
- Divide both sides by “2”: y = -4/2x + 9/2
- The slope is -4/2 or -2 .
- The opposite inverse of -2 is 1/2.
- Rearrange the equation. b = y – mx.
- Then enter the tournament.
- b = -1 – (1/2)8.
- b = -1 – (4).
- b = -5.
- Double check to make sure that -5 is exactly the origin.
- Write the equation: y = 1/2x – 5
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This article is co-authored by a team of editors and trained researchers who confirm the accuracy and completeness of the article.
The wikiHow Content Management team carefully monitors the work of editors to ensure that every article is up to a high standard of quality.
This article has been viewed 30,590 times.
To find the equation of a line, you need two things : a) a point on the line; and b) its slope (sometimes called its slope). But the way you find this information and what you can do with it later may vary from case to case. For simplicity, this article will focus on equations that take the form of slope and origin y = mx + b instead of the slope form and a point on the line (y – y 1 ) = m(x) – x 1 ) .
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