Question

Choose the equation in standard form of the line that passes through the point \((-1,-4)\) and has a slope of 2. (A) \(2 x+y=-2\) (B) \(-2 x+y=-2\) (C) \(2 x+y=2\) (D) \(2 x-y=-2\)

Short answer

The correct equation in standard form of the line that passes through the point (-1,-4) and has a slope of 2 is (D) \(2x - y = 2\).

Step-by-step solution

Use the Point-Slope Form

Start with the point-slope form of a line, given by \(y - y_1 = m(x - x_1)\), where \((x_1, y_1)\) is the point it passes through and \(m\) is the slope. Here, substitute \(-1\) for \(x_1\), \(-4\) for \(y_1\), and \(2\) for \(m\). When you do this, the equation becomes \(y + 4 = 2(x + 1)\).

Distribute the Slope

Spread out the right side by multiplying 2 with each term inside the parentheses. The equation becomes \(y + 4 = 2x + 2\).

Move \(2x\) to the left side

Alter the equation to standard form, \(Ax + By = C\), by moving \(2x\) to the left side. It yields \(-2x + y = -2\). By multiplying through by -1 gives \(2x - y = 2\).

Find the Correct Equation

After having derived the equation in standard form, find matching option from the given choices. The correct answer is (D) \(2x - y = 2\).

Key concepts

Point-Slope Form

The point-slope form is essential for writing the equation of a line when you are given a point on the line and the slope. The general formula is \( y - y_1 = m(x - x_1) \), where \( (x_1, y_1) \) represents the coordinates of the known point and \( m \) represents the slope of the line.

To grasp this concept, imagine you are plotting a graph and you have a starting point (the given point) and a direction in which to continue (the slope). With these two pieces of information, you can draw an entire line. This is exactly what the point-slope formula allows one to do algebraically. When given a specific point and slope, simply plug them into the formula to obtain an equation that models the line on a coordinate plane.

In the context of our exercise, we begin with the given point \( (-1,-4) \) and slope \( 2 \). By substituting these values into the point-slope formula, we derived the equation \( y + 4 = 2(x + 1) \) which is the first step towards finding the line's standard form equation.

Slope of a Line

Understanding the slope of a line is critical when dealing with linear equations. The slope is a measure of how steep a line is and the direction in which it tilts. Represented by the letter \( m \), the slope is calculated as the rise over run, which is the change in \( y \) (the vertical change) over the change in \( x \) (the horizontal change).

The slope can be positive, negative, zero, or undefined. A positive slope means the line goes upwards as it moves from left to right, while a negative slope means it goes downwards. A zero slope describes a horizontal line, indicating no vertical change at all. An undefined slope, on the other hand, corresponds to a vertical line, where there is no horizontal change.

In the exercise we are examining, the slope is given as \( 2 \). This is a positive integer, indicating the line rises two units vertically for every one unit it moves horizontally to the right. This relationship between vertical and horizontal movements is the foundational concept that establishes the directionality of the line on a graph.

Linear Equations

Linear equations form the backbone of coordinate geometry, particularly when it comes to graphing lines. These equations can be represented in different forms, including but not limited to point-slope form, slope-intercept form (\( y = mx + b \)), and the standard form (\( Ax + By = C \)). The standard form is particularly useful for analyzing the intercepts on the axes and for performing certain algebraic operations.

To write an equation in standard form, we aim to have \( A \) and \( B \) be integers, and the leading coefficient \( A \) should be a positive value. The most distinctive aspect of standard form is its organization, which places \( x \) and \( y \) on the same side, leading to a tidy presentation of the line's behavior on the graph.

During the exercise, starting from the point-slope form derived earlier, we manipulated the equation by distributing the slope and rearranging the terms to reach \( 2x + y = 2 \) which we then compared with the answer choices. Our objective was to properly format the equation to align with the standard form conventions, and in doing so, we successfully identified the correct equation from the given options.