Introduction to Straight Line Equations

Understanding the equation of a straight line is fundamental in both algebra and geometry. One common scenario is finding a line that intersects the y-axis at a specific point and is equally inclined to the coordinate axes. This article will guide you through the process of determining such a line's equation.

Understanding the Problem

The problem at hand involves finding the equation of a straight line that cuts the y-axis at a length of 5 units and is equally inclined to the coordinate axes. An equally inclined line has a slope of either 1 or -1.

Calculating the Y-Intercept

The first step involves identifying the y-intercept of the line. A line that cuts the y-axis at 5 units can be represented by the points (0, 5) and (0, -5). Since the y-intercept is the point where the line crosses the y-axis when x 0, we can use the point (0, 5).

Determining the Slope

An equally inclined line to the axes has a slope of 1 or -1. This can be easily derived from the fact that the line forms a 45-degree angle with both the x-axis and y-axis. Therefore, the line can either rise or fall at a 45-degree angle.

Formulating the Equation

Once we have the y-intercept and the slope, we can use the slope-intercept form of a line, which is:

y mx c

Where:

m is the slope of the line c is the y-intercept of the line

Substituting the values for m and c, we get two possible equations based on the slope:

For m 1, the equation becomes:

y x 5

For m -1, the equation is:

y -x 5

Conclusion

Therefore, the equations for the line that cuts off an intercept of length 5 on the y-axis and is equally inclined to the axes are:

y x 5 (slope 1) y -x 5 (slope -1)

This method can be applied to similar problems where you need to find the equation of a line based on its intercept and inclination.

Additional Resources

Keywords: slope, y-intercept, equally inclined, straight line

For further reading: Explore more about the geometry of lines and their properties on Wikipedia.