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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
- 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).
- 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.
Know the slope and a point on the line
- 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.
- 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
- 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.
- 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.
- 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
- 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
- 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.
- 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
- 2/3 becomes -3/2
- -6/5 becomes 5/6
- 3 (or 3/1 — the same) becomes -1/3
- -1/2 becomes 2
- 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.
- 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
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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