Determining the Value of Y in a Mathematical Line Equation

In this article, we will focus on a specific problem that involves determining the value of y in a mathematical line equation, given that the gradient (or slope) of the line is 2, and that it passes through the points 35 and 5y. This type of problem is crucial in understanding the relationship between the gradient of a line and points it passes through, and it plays a significant role in the study of algebra and geometry.

Understanding the Problem

When solving such problems, we need to recall the fundamental definition of the gradient or slope of a line, which is given by:

The gradient $m frac{Delta y}{Delta x}$

where Δy is the change in the y coordinate and Δx is the change in the x coordinate.

Solving the Given Problem

Let's consider the problem in detail: we are given that the gradient of a line passing through the points 35 and 5y is 2. We need to find the value of y.

Step-by-Step Solution

According to the gradient formula, we have:

$frac{y - 5}{5 - 3} 2$

Next, we simplify the expression:

$frac{y - 5}{2} 2$

Then, we cross-multiply to solve for y:

$y - 5 4$

Adding 5 to both sides of the equation, we get:

$y 9$

Therefore, the value of y is 9. This solution is verified through the substitution back into the original equation, ensuring the accuracy of our result.

Additional Information and Context

Furthermore, this problem can be extended to various other forms of linear equations. For example, given a line in the form $y 2x - 1$, if we want to find the value of y when $x 5$, we can simply substitute 5 for x in the equation:

When $x 5$, $y 2(5) - 1 10 - 1 9$

Similarly, if the equation is written in the form $y 2x b$ and we know that when $x 3$, $y 5$, we can solve for b and then find y when $x 5$.

Conclusion

Understanding and solving problems like this not only helps in mastering the concept of gradients and slopes but also enhances problem-solving skills in algebra and geometry. It is essential for students and professionals in mathematics and related fields to have a firm grasp of these basic principles.

Keywords

- Gradient - Mathematical Line Equation - Slope Calculation