Question

Write the point-slope form of the line with slope 5 containing the point (3,-2)

Short answer

y + 2 = 5(x - 3)

Step-by-step solution

Identify the key components

Find out the slope and the coordinates of the point on the line. Here, the slope is given as 5 and the point is (3, -2).

Recall the point-slope formula

The point-slope form of a line is: \(y - y_1 = m(x - x_1)\). Here, \(m\) is the slope, and \( (x_1, y_1) \) are the coordinates of the point.

Substitute the given values

Substitute the slope \( m = 5 \), \( x_1 = 3 \), and \( y_1 = -2 \) into the point-slope formula: \((y - (-2) = 5(x - 3))\).

Simplify the equation

Simplify the equation from the previous step: \(y + 2 = 5(x - 3)\).

Key concepts

slope

The slope of a line measures its steepness. It tells us how much the y-coordinate changes when the x-coordinate increases by 1. The formula for the slope, often denoted as 'm', between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) on a line is given by: \[ m = \frac{(y_2 - y_1)}{(x_2 - x_1)} \]. In this exercise, the slope is already provided as 5, meaning for every unit increase in x, y increases by 5 units.

coordinates

Coordinates are pairs of numbers that define the position of a point in a plane. They are written as (x, y), where x represents the horizontal position and y represents the vertical position. In the given exercise, the point on the line is (3, -2). Here, 3 is the x-coordinate and -2 is the y-coordinate. These coordinates help us pinpoint the exact location on the graph where the line passes through.

equation of a line

The equation of a line describes all the points that lie on the line. There are several forms of linear equations, including the slope-intercept form \( y = mx + b \) and the standard form \( Ax + By = C \). In this exercise, we are focusing on the point-slope form of the line, which is very useful when we know a point on the line and the line's slope.

simplification

Simplification is the process of making an equation or expression easier to work with. In our exercise, after substituting the given values into the point-slope formula, we simplify the equation by combining like terms and removing unnecessary parentheses to make it more readable. Simplifying \( y - (-2) = 5(x - 3) \) results in \( y + 2 = 5(x - 3) \). This form is more straightforward and easier to understand.

point-slope formula

The point-slope formula is written as \( y - y_1 = m(x - x_1) \). It is a valuable tool for writing the equation of a line when you know the slope and a point on the line. In our exercise, the given slope is 5 and the given point is (3, -2). Substituting these values into the formula, we get \( y - (-2) = 5(x - 3) \). Simplifying it, we arrive at \( y + 2 = 5(x - 3) \). This formula directly ties the slope (rate of change) and a specific point on the line to the overall equation of the line.