Question

Write an equation of the line that passes through the points. $$(1,-4),(2,-1)$$

Short answer

The equation of the line that passes through the points (1, -4) and (2, -1) is \(y = 3x - 7\).

Step-by-step solution

Calculate the slope

Use the formula for the slope which is \(m = \frac{(y2 - y1)}{(x2 - x1)}\). Substitute the given coordinates into the formula, so it becomes \(m = \frac{(-1 - (-4))}{(2 - 1)}\) which simplifies to \(m = 3\).

Write the equation

Use the point-slope form \(y - y1 = m(x - x1)\) to write the equation. Substitute m = 3 and use the point (1,-4) for (x1, y1) to get the equation \(y - (-4) = 3(x - 1)\). This simplifies to \(y + 4 = 3x - 3\). After rearranging the numbers, the final equation is \(y = 3x - 7\).

Key concepts

Slope Calculation

Calculating the slope is an essential first step in understanding linear equations. The slope of a line measures its steepness and direction. You can determine the slope by using two points on the line. Here's the formula you use:

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

In this formula, \( m \) stands for the slope, \((x_1, y_1)\) and \((x_2, y_2)\) represent the coordinates of the two points. Insert the values from the points given in the exercise:

• \( x_1 = 1, y_1 = -4 \)
• \( x_2 = 2, y_2 = -1 \)

Substitute these into the formula:

• \( m = \frac{-1 - (-4)}{2 - 1} \)
• \( m = \frac{3}{1} \)

Thus, the slope \( m \) is 3, meaning for every unit you move the line horizontally, it moves three units vertically.

Point-Slope Form

The point-slope form of a linear equation is a crucial tool when you need to find the equation of a line. It's particularly helpful if you know the slope and a single point on the line. The general representation of this form is:

• \( y - y_1 = m(x - x_1) \)

In this equation:

• \((x_1, y_1)\) are the coordinates of the known point
• \(m\) is the slope of the line

Using the point (1, -4) and the slope \( m = 3 \), plug these values into the formula:

• \( y - (-4) = 3(x - 1) \)
• This simplifies to \( y + 4 = 3x - 3 \)

Now, you have a point-slope equation, which describes the relationship between \( y \) and \( x \) based on the slope and the known point.

Equation of a Line

Once you find the slope and understand the point-slope form, you can convert this into the standard equation of a line, which usually appears as \(y = mx + b\). The goal here is to express the equation in a more conventional form that is easier to use in various applications.From the point-slope form we derived:

• \( y + 4 = 3x - 3 \)

You rearrange this to the standard form:

• Subtract 4 from both sides: \( y = 3x - 3 - 4 \)
• This simplifies to \( y = 3x - 7 \)

Now, \(y = 3x - 7\) represents the line's equation in its slope-intercept form, where 3 is the slope, and -7 is the y-intercept. This expression shows how \(y\) changes with \(x\), and serves as a tool for graphing or analyzing the line's behavior.