Understanding the Equation of a Straight Line Through a Given Point and Y-Intercept

Introduction

When dealing with the equation of a straight line in linear algebra, you often come across situations where you need to find the equation of a line given specific points, particularly when one of those points is on the y-axis. This article will guide you through the process of determining the equation of a straight line passing through a given point (1, 2) and having a y-intercept of 3.

Step-by-Step Guide to Finding the Equation

The equation of a straight line can be expressed in a general form as follows:

y mx c

Where:

m is the slope of the line. c is the y-intercept of the line, which is the value of y when x is zero.

Step 1: Identifying the Y-Intercept

The problem states explicitly that the y-intercept is 3. Therefore, the value of c is 3.

Step 2: Calculating the Slope

To calculate the slope, we can use the formula for the slope between two points. Given the point (1, 2) and the y-intercept (0, 3), we can find m as follows:

m (y_2 - y_1) / (x_2 - x_1)

Substituting the coordinates of the given point and the y-intercept:

m (3 - 2) / (0 - 1) 1 / -1 -1

Step 3: Writing the Equation of the Line

Now that we have the slope m -1 and the y-intercept c 3, we can write the equation of the line:

y -x 3

This is the required equation of the straight line.

Conclusion

The equation of the straight line that passes through the point (1, 2) and has a y-intercept of 3 is:

y - x 3

This method is widely applicable and can be used to find the equation of a line when the y-intercept and another point on the line are provided.

Further Reading and Resources

For a deeper understanding and additional practice, you may want to explore the following resources:

Linear Equations on MathIsFun Khan Academy: Slope-Intercept Form

Through practice and exploration, you will become more confident in handling various scenarios involving the equation of a straight line.