Problem 33 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ y+4 x=0 $$

Step-by-Step Solution

Answer

Graph the line starting at (0,0) using slope -4, passing through (1,-4).

Step 1: Write the Equation in Slope-Intercept Form

To better understand how to graph the equation, we first need to write it in the form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. Starting with the given equation \( y + 4x = 0 \), solve for \( y \) by subtracting \( 4x \) from both sides to get \( y = -4x \).

Step 2: Identify the Slope and Y-Intercept

In the equation \( y = -4x \), we can see that the slope \( m \) is \(-4\) and the y-intercept \( b \) is 0. This tells us the line passes through the origin at (0,0) and has a slope of \(-4\), which means it moves down 4 units vertically for every 1 unit it moves horizontally to the right.

Step 3: Plot the Y-Intercept

Begin graphing by plotting the y-intercept point on the graph. For this equation, the y-intercept is at (0,0), where the line crosses the y-axis.

Step 4: Use the Slope to Plot Another Point

Starting from the y-intercept (0,0), use the slope \(-4\) to find another point. Since the slope is \(-4\), go down 4 units and 1 unit to the right from (0,0) to place the next point at (1, -4).

Step 5: Draw the Line

Connect the two plotted points (0,0) and (1,-4) with a straight line. This line represents the graph of the equation \( y = -4x \). Extend the line across the coordinate plane, making sure it continues in both directions.

Key Concepts

Slope-Intercept Form

When graphing linear equations, it's essential to understand the slope-intercept form, which is defined as: \[ y = mx + b \] Here, \( m \) represents the slope of the line, while \( b \) is the y-intercept. This form makes it easier to quickly identify key characteristics of the line on a graph. The slope \( m \) indicates how steep the line is and in which direction it slants. A positive slope means the line angles upwards from left to right, while a negative slope angles downwards. The y-intercept \( b \) points to where the line crosses the y-axis. Thus, the slope-intercept form provides critical insights for graphing a linear equation directly onto a coordinate plane.

Linear Equation Graphing

Graphing a linear equation like \( y = -4x \) involves a few straightforward steps. First, rearrange the given equation into the slope-intercept form, which was shown to be \( y = -4x \) in this exercise. In this form, the slope \( m \) is \(-4\), and the y-intercept \( b \) is \(0\). To start graphing, plot the y-intercept point, which is \((0, 0)\) for our equation. Next, use the slope \(-4\) as a guide to find the second point. This negative slope tells us the line drops 4 units vertically for every unit it moves horizontally to the right. From \((0, 0)\), move to \((1, -4)\) and mark this point. Once you have two points, draw a straight line connecting them. Extend the line across the graph, demonstrating the continuous nature of the linear relationship defined by the equation.

Coordinate Plane

A coordinate plane is a two-dimensional surface on which we can plot points, lines, and curves to visually represent mathematical relationships. It consists of two axes: a horizontal x-axis and a vertical y-axis, intersecting at a point called the origin \((0, 0)\). When graphing equations like \( y = -4x \), the coordinate plane allows you to see the direction and steepness of the line. Points are plotted based on their \( (x, y) \) coordinates. The axes divide the plane into four quadrants, and understanding which quadrant a line passes through can give insights about its slope and intercepts.

• The first quadrant consists of positive x and y values.
• The second quadrant has negative x and positive y values.
• The third quadrant includes both negative x and y values.
• The fourth quadrant contains positive x and negative y values.

In our exercise, the line for \( y = -4x \) starts from the origin and moves into the fourth quadrant due to its negative slope, creating a visual representation of the relationship defined by the equation.