Finding The Value Of V Given Gradient And Two Points
In mathematics, particularly in coordinate geometry, understanding the relationship between points, lines, and gradients is fundamental. This article delves into a specific problem: determining the value of v given two points and the gradient of the line passing through them. The problem is stated as follows: the straight line that passes through the points (4, v) and (8,1) has a gradient of 5/7. The objective is to find the value of v and express it as an integer or a fraction in its simplest form.
Understanding Gradient
Before diving into the solution, let's revisit the concept of the gradient. The gradient, often denoted as m, is a measure of the steepness of a line. It quantifies how much the line rises (or falls) for every unit increase in the horizontal direction. Mathematically, the gradient between two points (x₁, y₁) and (x₂, y₂) is calculated using the formula:
m = (y₂ - y₁) / (x₂ - x₁)
This formula represents the change in the vertical coordinate (y) divided by the change in the horizontal coordinate (x). A positive gradient indicates an upward slope, while a negative gradient indicates a downward slope. A gradient of zero represents a horizontal line, and an undefined gradient represents a vertical line.
In the given problem, we are provided with the gradient (5/7) and two points (4, v) and (8,1). Our task is to use this information and the gradient formula to solve for the unknown value, v.
Applying the Gradient Formula
To find the value of v, we will substitute the given values into the gradient formula. Let's assign the points as follows:
(x₁, y₁) = (4, v) (x₂, y₂) = (8, 1)
The gradient, m, is given as 5/7. Now, we can plug these values into the gradient formula:
5/7 = (1 - v) / (8 - 4)
This equation relates the gradient to the coordinates of the two points. The next step is to simplify and solve for v.
Solving for v
First, simplify the denominator on the right side of the equation:
5/7 = (1 - v) / 4
To isolate the term containing v, multiply both sides of the equation by 4:
(5/7) * 4 = 1 - v
This simplifies to:
20/7 = 1 - v
Now, to solve for v, we need to isolate it on one side of the equation. Add v to both sides and subtract 20/7 from both sides:
v = 1 - 20/7
To combine the terms on the right side, we need a common denominator. Convert 1 to 7/7:
v = 7/7 - 20/7
Now, subtract the fractions:
v = -13/7
Therefore, the value of v is -13/7. This is a fraction in its simplest form, as 13 and 7 are both prime numbers and have no common factors other than 1.
Verification
To ensure the correctness of our solution, we can substitute the value of v back into the gradient formula and check if it yields the given gradient of 5/7. Substituting v = -13/7 into the gradient formula, we get:
m = (1 - (-13/7)) / (8 - 4)
Simplify the numerator:
m = (1 + 13/7) / 4
Convert 1 to 7/7 and add the fractions:
m = (7/7 + 13/7) / 4
m = (20/7) / 4
Divide by 4 (which is the same as multiplying by 1/4):
m = (20/7) * (1/4)
m = 20/28
Simplify the fraction by dividing both numerator and denominator by their greatest common divisor, which is 4:
m = 5/7
This confirms that our calculated value of v = -13/7 is correct, as it yields the given gradient of 5/7 when substituted back into the gradient formula.
Conclusion
In this article, we successfully determined the value of v given two points and the gradient of the line passing through them. We began by revisiting the concept of the gradient and its formula. Then, we applied the gradient formula to the given points and gradient, set up an equation, and solved for v. The solution was found to be v = -13/7. Finally, we verified our solution by substituting it back into the gradient formula and confirming that it yields the given gradient. This problem highlights the importance of understanding and applying the gradient formula in coordinate geometry. The ability to calculate gradients and use them to solve for unknown coordinates is a crucial skill in mathematics and has applications in various fields, including physics, engineering, and computer graphics.
Linear equations and gradients are fundamental concepts in mathematics, particularly in algebra and coordinate geometry. A linear equation represents a straight line on a coordinate plane, and the gradient describes the steepness and direction of that line. Understanding these concepts is crucial for solving a wide range of problems, from simple algebraic equations to more complex applications in physics and engineering. This article will explore the relationship between linear equations, gradients, and how to find unknown values using these principles. We will delve into a specific problem where we need to determine the value of a variable given two points on a line and its gradient. This problem will serve as a practical example of how these concepts are applied.
The Basics of Linear Equations
A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. The general form of a linear equation in two variables, x and y, is:
y = mx + c
where:
- y is the dependent variable (usually plotted on the vertical axis)
- x is the independent variable (usually plotted on the horizontal axis)
- m is the gradient of the line
- c is the y-intercept (the point where the line crosses the y-axis)
The gradient (m) is a measure of the steepness of the line. It represents the change in y for a unit change in x. A positive gradient indicates that the line slopes upwards from left to right, while a negative gradient indicates that the line slopes downwards. A gradient of zero means the line is horizontal. The y-intercept (c) is the value of y when x is zero, and it determines where the line crosses the vertical axis.
Understanding the components of a linear equation is essential for analyzing and interpreting the behavior of straight lines on a graph. The gradient and y-intercept provide crucial information about the line's orientation and position in the coordinate plane.
Calculating the Gradient
The gradient of a line can be calculated if we know two points on the line. Let's say we have two points (x₁, y₁) and (x₂, y₂). The gradient (m) can be calculated using the following formula:
m = (y₂ - y₁) / (x₂ - x₁)
This formula represents the change in the y-coordinates divided by the change in the x-coordinates. It essentially measures the
