Problem 41 - Algebra 1: Concepts and Skills

Question

Write in slope-intercept form the equation of the line that passes through the given point and has the given slope. $$ (-4,-1), m=-2 $$

Step-by-Step Solution

Answer

The equation of the line that passes through the given point (-4,-1) and has a slope of -2 in slope-intercept form is \(y = -2x - 9\).

Step 1: Substituting for \(m\), \(x\), and \(y\) in the slope-intercept equation.

First, substitute the provided slope \(m = -2\) and coordinates \((x, y) = (-4, -1)\) into the equation \(y = mx + b\). This yields:\n\(-1 = -2(-4) + b\)

Step 2: Solving for \(b\) the y-intercept.

Solve the derived equation to find the y-intercept \(b\). Hence, \(-1 = 8 + b\). After rearranging, one gets: \(b = -1 - 8 = -9\)

Step 3: Writing down the final answer.

Substitute \(m = -2\) and \(b = -9\) into the equation \(y = mx + b\) to get the equation of the line in slope-intercept form, which is \(y = -2x - 9\)

Key Concepts

Linear Equation

A linear equation is a simple yet powerful mathematical tool that represents a straight line on a graph. The general form of a linear equation is given by \( y = mx + b \), where \( m \) stands for the slope of the line, and \( b \) represents the y-intercept. The slope \( m \) indicates how steep the line is, reflecting the change in \( y \) with respect to the change in \( x \).
In the context of the given problem, the slope is \( m = -2 \). This means for every unit increase in \( x \), the value of \( y \) decreases by 2 units due to the negative sign. Understanding this simple relationship can help visualize how the linear equation behaves when plotted on a graph.

• The slope is crucial as it determines the direction and angle of the line.
• Every linear equation graphically shows a straight line.

If you can determine the slope and a point the line passes through, you can write the equation in slope-intercept form effectively.

Y-intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form \( y = mx + b \), the \( b \) denotes this specific point. The y-intercept gives the starting point of the line when \( x = 0 \). This is a key point in understanding and plotting the linear equation on a coordinate plane.
Determining the y-intercept, as seen in the solution, involves substituting the known slope \( m \) and a point \((x, y)\) through which the line passes to solve for \( b \). Here, the identified y-intercept is \( b = -9 \), which means the line crosses the y-axis at the point \((0, -9)\).

• The y-intercept provides a visualization cue of where the line is placed vertically on the graph.
• Knowing the y-intercept helps in graphing the linear equation starting with a known point.

Always check the calculated y-intercept for accuracy to ensure the line's proper depiction when plotting on the graph.

Coordinate Geometry

Coordinate geometry is the branch of mathematics that lets us represent geometric figures and relationships using a coordinate plane. It's a practical framework that connects algebra with geometry.
In coordinate geometry, we work within a two-dimensional plane where each point is identified by an ordered pair of numbers, \((x, y)\). The x-coordinate indicates the position along the horizontal axis, whereas the y-coordinate sets the point's position along the vertical axis.

• Points, lines, and other geometric shapes can be represented and analyzed using coordinates.
• Solving geometric problems becomes straightforward with equations like the linear equation.

For the specific exercise, the line passing through the point \((-4, -1)\) with a slope of \(-2\) is represented by the equation \( y = -2x - 9 \). In coordinate geometry, understanding how points and lines relate through this kind of representation allows for precise graphical plotting and solving complex geometry problems seamlessly.