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Solving for y: Finding the Intersection of the Gradient and Line
Introduction to Gradient and Line Equations
In mathematics, particularly in algebra and geometry, the gradient (or slope) of a line is a fundamental concept used to describe the change in y-values with respect to x-values. This concept is often applied to solve real-world problems, such as finding the point of intersection of two lines. This article will guide you through a detailed process of solving for y in the line that passes through the points (-1, y) and (2, 8) with a gradient of 2.
Understanding the Problem
The problem is to find the value of y such that the line joining the points (-1, y) and (2, 8) has a gradient of 2.
Gradient Calculations
The gradient m of a line joining two points (x1, y1) and (x2, y2) is given by:
m (y2 - y1) / (x2 - x1)
Example 1
Let's use the given values: (x1, y1) (-1, y) and (x2, y2) (2, 8). The gradient m 2.
Substituting these values into the gradient formula:
m (8 - y) / (2 - (-1)) 2
m (8 - y) / 3 2
Multiplying both sides by 3:
8 - y 6
Rearranging to solve for y:
-y 6 - 8
-y -2
y 2
Example 2
Another way to solve for y is by using the gradient formula:
m (y2 - y1) / (x2 - x1)
m (8 - y) / (2 - (-1)) 2
m (8 - y) / 3 2
Multiplying both sides by 3:
8 - y 6
Rearranging to solve for y:
-y 6 - 8
-y -2
y 2
Example 3
Using the same gradient formula:
8 - y / 3 2
Multiplying both sides by 3:
8 - y 6
Rearranging to solve for y:
-y 6 - 8
-y -2
y 2
Example 4
Let's also explore a less conventional method that involves cross-multiplication:
(8 - y) / (2 - (-1)) 2
Multiplying both sides by the denominator:
(8 - y) 2 * (2 - (-1))
8 - y 2 * 3
8 - y 6
Rearranging to solve for y:
-y 6 - 8
-y -2
y 2
Conclusion
In conclusion, the value of y that makes the line joining the points (-1, y) and (2, 8) have a gradient of 2 is 2. Proper understanding and application of the gradient formula are essential in solving such problems, making the algebraic process more straightforward.
Keywords: gradient, line equation, algebra