Finding the Equation of a Straight Line Given Specific Intercepts

When dealing with the equation of a straight line, determining the intercepts (x-intercept and y-intercept) can offer valuable insight. This article illustrates how to find the equation of a straight line that passes through a given point and has a y-intercept twice as long as its x-intercept. This process involves understanding and applying basic principles from algebra and coordinate geometry.

Understanding the Problem

Let's consider a scenario where we need to find the equation of a straight line that passes through the point (1, -3) and has a y-intercept twice as long as its x-intercept. This problem requires us to use our knowledge of intercepts and algebraic manipulation to arrive at the correct equation.

Step-by-Step Solution

To solve this problem, follow these steps:

Determine the Intercepts: Let the x-intercept be a. This means the line crosses the x-axis at the point (a, 0). Since the y-intercept is twice the x-intercept, we denote the y-intercept as b 2a. Therefore, the line crosses the y-axis at the point (0, 2a). Form the Line Equation: Using the two intercepts, the intercept form of the line can be written as: [ frac{x}{a} frac{y}{2a} 1 ]

Multiplying through by 2a gives:

[ y -frac{1}{2}x a ]

Substituting the point (1, -3) into the equation to solve for a will give us the specific line:

[ -3 -frac{1}{2}(1) a ]

Rewriting the equation:

[ -3 -frac{1}{2} a ]

Adding (frac{1}{2}) to both sides:

[ a -3 frac{1}{2} -frac{6}{2} frac{1}{2} -frac{5}{2} ]

Now substituting a back into the line equation:

[ y -frac{1}{2}x - frac{5}{2} ]

To express this in standard form, we can multiply through by 2:

[ 2y -x - 5 ]

Rearranging gives:

[ x 2y 5 0 ]

Thus, the equation of the line is:

Equation of the Line: ( x 2y 5 0 )

Exploring a Different Approach

Consider another scenario where the ratio of the x-intercept to the y-intercept is 2:3. In this case, the slope of the line must be (-frac{3}{2}). However, let's verify if a line with this slope passing through the point (-1, 3) adheres to the given ratio of intercepts.

Start by assuming the slope m of the line is (-frac{3}{2}) and the line passes through the point (-1, 3).

The point-slope form of the line can be written as:

[ y - 3 -frac{3}{2}(x 1) ]

This can be converted to the slope-intercept form:

[ y -frac{3}{2}x - frac{3}{2} 3 ]

Simplifying further:

[ y -frac{3}{2}x frac{3}{2} ]

Multiplying through by 2 to eliminate the fraction:

[ 2y -3x 3 ]

Rearranging to standard form:

[ 3x 2y - 3 0 ]

Given that the ratio of intercepts is 2:3, the slope must be -(frac{3}{2}) for the line to traverse the correct quadrant.

Conclusion

Through these examples, we've seen how to find the equation of a straight line given specific intercepts. The key steps involve understanding the intercept form of the line, substituting the given point, and solving for the unknown intercepts. This approach can be applied to a variety of linear equations, making it a valuable skill in algebra and coordinate geometry.