Math Lesson 14.3.3 - Finding the Equation of a Line that Passes through Two Known Points

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Welcome to our Math lesson on Finding the Equation of a Line that Passes through Two Known Points, this is the third lesson of our suite of math lessons covering the topic of Equation of Linear Graphs, you can find links to the other lessons within this tutorial and access additional Math learning resources below this lesson.

Finding the Equation of a Line that Passes through Two Known Points

Now, let's consider the reverse problem, i.e. when the linear equation is unknown but we know two points of the line. These two points, say A(x₁, y₁) and B(x₂, y₂) help us find the equation of the line expressed in the form y = mx + n in two steps:

Step 1: Calculating the gradient m through the formula

Step 2: Substitute the value found in Step 1 and the coordinates of one of the known points in the equation of the line to find the constant c.

Let's explain this procedure through a couple of examples.

Example 4

Find the equation of the straight line that passes through the points

1. A(-1, 2) and B(3, 4)
2. A(2, -4) and B(5, -3)

Solution 4

1. First, let's calculate the gradient m. Thus, from the equation of gradient
   m = y_B - y_A/x_B - x_A
   we obtain after substituting the known values:
   m = 4 - 2/3 - (-1)
   = 4 - 2/3 + 1
   = 2/4
   = 1/2
   Now, we substitute the coordinates of any of the two given points (for example those of point B given that both values are positive) in the equation
   y = mx + n
   which already became
   y = 1/2 x + n
   This allows us to find the constant n. Thus,
   4 = 1/2 ∙ 3 + n
   4 = 3/2 + n
   n = 4 - 3/2
   = 8/2 - 3/2
   = 5/2
   Therefore, the line's equation that passes through the two given points is
   y = 1/2 x + 5/2
2. The procedure is the same as in (a). Thus, first, we find the gradient m,
   m = y_B - y_A/x_B - x_A
   = -3 - (-4)/5 - 2
   = -3 + 4/3
   = 1/3
   then, we find the constant n by substituting the coordinates of any point in the line's equation. For example let's choose the coordinates of point A(2, -4) to insert into the equation
   y = 1/3 x + n
   This allows us to find the constant n. Thus,
   -4 = 1/3 ∙ 2 + n
   -4 = 2/3 + n
   n = -4 - 2/3
   = -12/3 - 2/3
   = -14/3
   Hence, the equation of the line that passes through the two given points is
   y = 1/3 x - 14/3

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