Question

Write an equation in point-slope form of the line that passes through the given point and has the given slope. \((3,2) ; m=-2\)

Short answer

The equation of the line in point-slope form that passes through the point \((3,2)\) with slope -2 is \(y - 2 = -2(x - 3)\)

Step-by-step solution

Identify the given point and slope

The point given is \((3,2)\), so \(x_1 = 3\) and \(y_1 = 2\). The slope given, designated as \(m\), is -2.

Substitute the point coordinates and slope in point-slope form equation

Substitute \(x_1 = 3\), \(y_1 = 2\), and \(m = -2\) into the point-slope form equation \(y - y_1 = m(x - x_1)\), we obtain: \[y - 2 = -2(x - 3)\]

Key concepts

Linear Equations

Linear equations are the backbone of algebra, serving as a simple yet powerful tool to model relationships between variables. The point-slope form is one way to express a linear equation. It's particularly useful because it doesn't require you to know the y-intercept right away.

Here's how it works: with point-slope form, represented by the formula \( y - y_1 = m(x - x_1) \). This equation lets you describe a line by using any point on the line \((x_1, y_1)\) and the slope \(m\).

The slope \(m\) indicates the steepness and direction of the line. A positive slope means the line rises as you move from left to right, while a negative slope means it falls. In summary, when you have a point and a slope, you can craft a linear equation that models the line they describe.

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, blends algebra with geometry by examining figures using a coordinate system. It allows us to study points, lines, and shapes using algebraic expressions.

In coordinate geometry, every point is identified by a pair of numerical coordinates \((x, y)\). These coordinates are derived from their distances to the x (horizontal) and y (vertical) axes on a Cartesian plane.

For example, in the problem at hand, the given point is \((3,2)\). This means 3 units along the x-axis and 2 units along the y-axis. Using such specific points makes visualizing and calculating geometric problems more structured and precise, especially when determining the equations of lines using specific points and slopes.

Algebra

Algebra is a branch of mathematics that uses symbols and letters to represent numbers and quantities in formulas and equations. It forms the language through which most mathematical concepts are communicated.

In the context of point-slope form, algebra is employed to manipulate and solve equations describing lines. For instance, once you've substituted a point and a slope into the point-slope equation, algebraic manipulation helps to simplify or rearrange the equation to other forms, like the slope-intercept or standard form.

Moreover, understanding how to rearrange equations and solve for variables is a key algebraic skill. Practicing with algebraic equations, like those in point-slope form, enhances problem-solving abilities and is essential for more advanced mathematical studies.