How to Determine the X-Intercept of a Line Passing Through Specific Points

Understanding how to determine the x-intercept of a line that passes through specific points is a fundamental skill in algebra and geometry. This guide will walk you through the process step-by-step, using the points (1, 7) and (2, 10) as examples. We'll cover finding the slope, deriving the line equation, and then solving for the x-intercept.

1. Finding the Slope (m)

The slope (m) of a line passing through two points can be calculated using the formula:

[ m frac{y_2 - y_1}{x_2 - x_1} ]

For the points (1, 7) and (2, 10), where (x_1, y_1) (1, 7) and (x_2, y_2) (2, 10), we can substitute these values into the formula:

[ m frac{10 - 7}{2 - 1} frac{3}{1} 3 ]

2. Deriving the Line Equation Using the Point-Slope Form

Once we have the slope, we can use the point-slope form to derive the equation of the line. The point-slope form is:

[ y - y_1 m(x - x_1) ]

Using the point (1, 7) and the slope ( m 3 ), the equation becomes:

[ y - 7 3(x - 1) ]

Simplifying this equation, we get:

[ y - 7 3x - 3 ]

[ y 3x - 3 7 ]

[ y 3x 4 ]

3. Finding the X-Intercept

The x-intercept is the point where the line crosses the x-axis, which means the value of ( y ) is 0. We can find this by setting ( y 0 ) in the line equation:

[ 0 3x 4 ]

Solving for ( x ), we get:

[ 3x -4 ]

[ x -frac{4}{3} ]

Therefore, the x-intercept of the line is at the point ( left(-frac{4}{3}, 0right) ).

Additional Methods

Another approach to finding the x-intercept is to use the point-point form of the line equation:

[ c - ay - b d - bx - a ]

For the points (7,1) and (10,2), we can substitute the values as follows:

[ 10 - 7y - 7 2 - 1x - 1 ]

Setting ( y 0 ) to find the x-intercept:

[ 10 - 7 2 - x - 1 ]

[ 3 2 - x - 1 ]

[ 3 1 - x ]

[ x -2 ]

However, using the slope formula and point-slope form is a more general and straightforward approach.

Conclusion

Determining the x-intercept is a crucial skill in algebra and can be applied to various real-world problems, such as calculating break-even points in business or understanding linear relationships in data analysis. By following the steps outlined in this guide, you can easily find the x-intercept of any line passing through two given points.