Understanding the Y-Intersection of a Line with the Equation -2x y - 7 0

In the realm of linear algebra and coordinate geometry, identifying the point at which a line intersects the y-axis is a fundamental concept. This process involves setting the x-coordinate of the point to zero and solving for the y-coordinate. The given line equation is -2x y - 7 0. This article will guide you through the steps to find this specific y-intercept.

The Concept of Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis in a coordinate system. This occurs when the x-coordinate is zero. For the line represented by the equation -2x y - 7 0, the x-coordinate of the y-intercept is 0.

Steps to Find the Y-Intercept

Substitute x 0 into the equation: -2(0) y - 7 0 Simplify the equation: 0 y - 7 0 Solve for y: y - 7 0 Therefore, y 7

Interpretation

The line -2x y - 7 0 intersects the y-axis at the point (0, 7). This means that when x is 0, the value of y is 7. So, the y-intercept is 7.

Relating to Other Forms of Line Equations

The line equation in the standard form can also be related to the intercept form. The intercept form of a line is given by (frac{x}{a} frac{y}{b} 1), where (a) is the x-intercept and (b) is the y-intercept. However, our equation is in the slope-intercept form (y mx c), where (c) is the y-intercept.

Given the equation -2x y - 7 0, converting it to the slope-intercept form gives us:

y 2x 7

Here, 7 is the y-intercept. This confirms that the y-intercept of the line -2x y - 7 0 is indeed 7, as we calculated earlier.

Conclusion

In summary, finding the y-intercept of a line involves substituting x 0 into the equation and solving for y. For the equation -2x y - 7 0:

Substitute x 0: -2(0) y - 7 0 Simplify: 0 y - 7 0 Solve: y 7 Hence, the y-intercept is 7

The y-intercept is the point (0, 7), which is the point where the line intersects the y-axis.