Question

Write an equation of the line that passes through the points. $$(2,-1),(0,1)$$

Short answer

The equation of the line that passes through the points (2, -1) and (0, 1) is \(y = -x + 1\)

Step-by-step solution

Calculate the slope

First, find the slope of the line that passes through the points using the formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Here, \((x_1, y_1) = (2, -1)\) and \((x_2, y_2) = (0, 1)\). Hence, the slope \(m = \frac{1 - (-1)}{0 - 2} = -1\).

Substitute the slope and one point into the equation of the line

With the slope \(m=-1\) and one of the points, say \((x_1, y_1) = \((2, -1)\), substitute into the formula of the line, \(y - y_1 = m(x - x_1)\). This gives the equation of the line as \(y - (-1) = -1 * (x - 2)\).

Simplify the equation

Simplify \(y - (-1) = -1 * (x - 2)\) to get the equation in slope-intercept form (i.e., \(y = mx + b\)). First distribute the slope on the right side to get \(y + 1 = -x + 2\). Then subtract 1 from both sides to obtain the final equation of the line: \(y = -x + 1\).

Key concepts

Slope

The slope of a line is a measure of its steepness and direction. It's represented by the letter \( m \) in mathematical equations.
You can calculate the slope between two points \((x_1, y_1)\) and \((x_2, y_2)\) using the formula:

• \( m = \frac{y_2 - y_1}{x_2 - x_1} \)

This formula gives the rate at which \( y \) changes with respect to \( x \). A positive slope means the line rises as it moves from left to right. A negative slope means it falls. A slope of zero indicates a horizontal line, and an undefined slope indicates a vertical line.
In our original exercise, the slope between the points \((2, -1)\) and \((0, 1)\) is calculated as \( m = -1 \). This tells us the line is declining as it moves from the first point to the second.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a line, making the slope and the y-intercept easy to identify. It is expressed as:

• \( y = mx + b \)

Here, \( m \) represents the slope, and \( b \) is the y-intercept, the point where the line crosses the y-axis. This format makes it simple to graph a line and understand its behavior at a glance.
To convert the equation of a line from point-slope form to slope-intercept form, distribute and simplify as necessary.
In our exercise, after simplifying \( y - (-1) = -1(x - 2) \), we find \( y = -x + 1 \). This equation is in slope-intercept form, where the slope \( m = -1 \) and the y-intercept \( b = 1 \).

Coordinate Geometry

Coordinate geometry, also known as analytic geometry, connects algebra and geometry through graphs and equations. It allows us to use algebraic methods to solve geometric problems by using coordinates on a graph.
Key concepts in coordinate geometry include:

• Points, defined by coordinates \((x, y)\)
• Lines, which can be represented by linear equations
• Distance and midpoint formulas for measuring and locating points

In the context of our exercise, coordinate geometry allows us to find the equation of a line passing through two specific points. By determining the slope and expressing the relationship between \( x \) and \( y \) in equation form, we effectively bridge the gap between numerical data and spatial representation.