Introduction

When working with lines in mathematics, finding the equation of a line that passes through specific points is a fundamental exercise. This article will guide you through the process of finding the equation of a line in standard form given two points: (1, 4) and (-3, -9). We will use multiple methods, including the slope-intercept form and the directional vector approach, to arrive at the solution.

Method 1: Using Slope-Intercept Form

The first method involves using the slope-intercept form of a line. The standard slope-intercept form is:

y mx c

where m is the slope and c is the y-intercept.

Step 1: Find the Slope

The slope m can be calculated using the formula:

m (y2 - y1) / (x2 - x1)

Substituting the points (1, 4) and (-3, -9) into the formula gives:

m (-9 - 4) / (-3 - 1) 13 / 4

Step 2: Find the y-intercept

To find the y-intercept c, we can use the point-slope form of the equation:

y mx c

Using the point (1, 4) and the slope m 13/4, we get:

4 (13/4) * 1 c

c 4 - 13/4 16/4 - 13/4 3/4

Step 3: Write the Equation in Standard Form

Now that we have both the slope and the y-intercept, we can write the equation in the form:

y (13/4)x 3/4

Multiplying every term by 4 to clear the fractions, we get:

4y 13x 3

Or, rearranging to standard form:

13x - 4y -3

Method 2: Using Directional Vectors

Another approach is to use the directional vector of the line and the normal vector. The directional vector of the line passing through the points (1, 4) and (-3, -9) is given by:

AB (4, 13)

The normal vector to the line, which is perpendicular to the directional vector, is:

BA (13, -4)

Step 1: Write the Equation of the Line

The equation of the line in normal form is:

13x - 4y c

To find the constant c, we substitute one of the points. Using the point (1, 4), we get:

13(1) - 4(4) c

13 - 16 c

c -3

Step 2: Standard Form of the Equation

Substituting c -3 into the equation, we get:

13x - 4y -3

Conclusion

Thus, the equation of the line that passes through the points (1, 4) and (-3, -9) in standard form is:

13x - 4y -3

Both methods yield the same result, confirming the accuracy of our solution. This approach can be used for any two points, and understanding both methods provides a deeper insight into the properties of lines and equations.