Question

Find an equation of the line passing through the points. . \((6,-1),(-2,1)\)

Short answer

The equation of the line is \( y = -0.25x - 2.5 \)

Step-by-step solution

Find the Slope

Given the two points \((x_1, y_1) = (6, -1)\) and \((x_2, y_2) = (-2, 1)\), we calculate the slope \(m\) using the slope formula \(m = \frac{y_2 - y_1}{x_2 - x_1}\). Substituting the given points into the formula gives:\[ m = \frac{1 - (-1)}{-2 - 6} = \frac{2}{-8} = -0.25 \]

Substitute the Slope and a Point into the Equation

Now we substitute the slope and either one of the points into the slope intercept form of a line \(y = mx + b\). Let's use the point \((6, -1)\). When we do this, it gives\[ -1 = -0.25(6) + b \]Solving this equation for \(b\), the y-intercept, gives us \(b = -0.25*6 + 1 = -2.5\).

Write Down the Equation of the Line

Now that we have the slope and the y-intercept, we can substitute these values into the slope-intercept form of a line equation \( y = mx + b \) to get the equation of the line passing through the given points. Therefore, the equation of the line passing through points \((6,-1)\) and \((-2,1)\) is \[ y = -0.25x - 2.5 \]

Key concepts

Slope-Intercept Form

Understanding the slope-intercept form is fundamental when dealing with linear equations. This form is written as: \[ y = mx + b \]where \( y \) is the dependent variable, \( x \) is the independent variable, \( m \) represents the slope of the line, and \( b \) denotes the y-intercept, which is the point where the line crosses the y-axis. To graph a line using this equation, one would start at the y-intercept (\( b \)) on the y-axis and use the slope (\( m \)) to determine the rise over run; that is, how many units to go up or down and how many to go right or left.

Calculating Slope

The slope of a line is a measure of its steepness and direction. Calculating the slope between two points requires the use of the formula: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \]In this formula, \( (x_1, y_1) \) and \( (x_2, y_2) \) are two distinct points on the line. The slope is essentially the ratio of the vertical change (the difference in the y-values) to the horizontal change (the difference in the x-values) between these two points. An upward sloping line has a positive slope, while a downward sloping line has a negative slope. A horizontal line has a slope of zero, and a vertical line's slope is undefined.

Linear Equations

Linear equations form the basis for a wide array of mathematical and real-world applications. These equations can always be written in the form of \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants. The solutions to these equations are an infinite set of points that, when plotted on a Cartesian plane, form a straight line.

When you encounter a linear equation, your goal is usually to solve for \( y \) in terms of \( x \) to get it into slope-intercept form, making it easier to understand and graph. Linear equations are used to describe relationships where there is constant rate of change between the variables.

Y-Intercept

The y-intercept is a specific point where a line crosses the y-axis on a graph. It's where the value of \( x \) is zero, so the y-intercept of any line can be found by setting \( x = 0 \) in the equation of the line.

In the slope-intercept form \( y = mx + b \), the y-intercept is represented by \( b \). This point is important as it gives a starting position for graphing the line and, along with the slope, completely determines the position of the line on the Cartesian plane. In real-world scenarios, the y-intercept is often considered the starting value or initial condition of a problem situation.