Question

The slope of the line joining the points \((2, \mathrm{k}-3)\) and \((4,-7)\) is 3 . Find \(\mathrm{k}\). (1) \(-10\) (2) \(-6\) (3) \(-2\) (4) 10

Short answer

Question: Determine the value of \(k\) if the line joining the points \((2, k-3)\) and \((4, -7)\) has a slope of 3. Answer: (4) 10

Step-by-step solution

Write down the slope formula

The slope formula is given by: \(m = \cfrac{y_2 - y_1}{x_2 - x_1} \)

Substitute the given points and slope

We are given the slope \(m = 3\), and the points \((x_1, y_1) = (2, k-3)\) and \((x_2, y_2) = (4, -7)\). Plug these values into the slope formula: \(3 = \cfrac{-7 - (k - 3)}{4 - 2}\)

Solve for k

Simplify and solve the equation for \(k\): \(3 = \cfrac{-7 - k + 3}{2}\) \(3(2) = -4 - k \) \(6 + 4 = k \) \(k = 10\) The value of \(k\) is 10, so the correct answer is (4) 10.

Key concepts

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, is the study of geometry using a coordinate system. This approach allows us to explore geometric shapes and their properties by translating them into algebraic equations that are much easier to analyze and solve. In coordinate geometry, points on a plane are defined by their coordinates, which are pairs of numbers corresponding to their position on the x-axis and y-axis.

In our exercise, we needed to find the slope of a line that passes through two points. One of the core principles of coordinate geometry is that the equation of a line in a two-dimensional space can be represented using its slope and y-intercept. When it comes to finding unknown variables linked to the geometry of lines, such as in our problem, it's essential to understand how these points interact within the grid system of the coordinate plane.

Understanding coordinate geometry is crucial for visualizing and solving problems involving lines, angles, and distances in the Cartesian plane. It is a fundamental part of higher mathematics and has wide applications in physics, engineering, computer science, and various other scientific fields.

Slope Formula

The slope of a line in coordinate geometry can be found using the slope formula, which is \(m = \cfrac{y_2 - y_1}{x_2 - x_1}\). The slope indicates how steep a line is and the direction it's going. A positive slope signifies an upward tilt from left to right, while a negative slope indicates a downward tilt.

When we refer to this formula, \(m\) represents the slope, whereas \(x_1, y_1\) and \(x_2, y_2\) represent the coordinates of two distinct points on the line. By subtracting the y-coordinates and dividing by the difference in x-coordinates, we find the rate at which the y-value changes per unit of x. This concept is pivotal when working with linear functions or analyzing how two variables might be related.

In our original exercise, we used the slope formula to connect the concept of slope with the given points. By plugging in the values into the formula, we translated a geometric concept (slope) into something we can mathematically solve, which is a prime example of how algebra and geometry are interlinked in coordinate geometry.

Algebraic Equations

Algebraic equations are mathematical statements that use numbers, variables, and arithmetic operations to express a relationship between quantities. The equation represents a balance, with both sides equal to each other. When you solve an algebraic equation, you're essentially finding the value(s) of the variable(s) that make the equation true.

In the context of our problem, we were presented with an algebraic equation once we substituted the known values into the slope formula. Solving this equation involved simplification steps and arithmetic operations that isolated the variable \(k\). Algebraic equations can range from simple, linear equations, to more complex polynomial equations, each requiring different techniques and methods for solving.

Our understanding of algebra is vital for solving a broad spectrum of problems in mathematics. Mastery of algebraic manipulation allows us to solve for unknowns within geometry problems, as we have seen with finding \(k\) using the slope formula. Algebra is the language through which we express and solve a wide array of mathematical problems.