Problem 42 - Algebra 1: Concepts and Skills

Question

Write the equation in slope-intercept form. Then graph the equation. $$ 6 x+y=0 $$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -6x\). The slope is \(-6\) and the y-intercept is \(0\). By starting at the y-intercept and using the slope to find another point, the line can be graphed.

Step 1: Rewrite equation into slope-intercept form

Start by moving the term containing \(x\) over to the other side of the equals sign. So, the equation \(6x + y = 0\) becomes \(y = -6x + 0\). This form of \(y = mx + b\) is slope-intercept form.

Step 2: Identify slope and y-intercept

In the equation \(y = -6x + 0\), the coefficient of \(x\), which is \(-6\), is the slope \(m\), and \(0\) is the y-intercept \(b\). Hence, the slope, \(m\), is \(-6\), and the y-intercept, \(b\), is \(0\).

Step 3: Graph the equation

Begin by plotting the y-intercept, which is \(0\), on the y-axis. After that, use the slope to determine the next point. The slope is \(-6\), which means for every unit increase in \(x\), \(y\) decreases by 6 units. So, from the y-intercept, if \(x\) moves right by 1 unit, \(y\) goes down by 6 units. Plot this point, and draw a line through the two points for the graph of the equation.

Key Concepts

Equation Graphing

Graphing equations is a fundamental skill in algebra, and the slope-intercept form makes this process straightforward. To graph an equation, you need to understand its visual representation on a coordinate plane. The slope-intercept form represents this most clearly and is written as \( y = mx + b \). Here, \( m \) is the slope of the line, and \( b \) is the y-intercept.

When graphing the equation, you start by plotting the y-intercept on the graph. This is the point where the line crosses the y-axis. From there, use the slope to find another point on the line. Drawing a line through these two points completes the graph.

• Start with the y-intercept: the point (0, b)
• Use the slope \( m \) to identify another point
• Draw a line through both points

This method helps visualize how changes in the equation affect the graph.

Slope of a Line

The slope of a line is an essential concept in understanding how steep a line is. It indicates how much \( y \) changes for a change in \( x \). In the equation \( y = mx + b \), the slope \( m \) informs us of the line's direction and steepness.

The slope can be interpreted from a rise over run perspective. If the slope is positive, the line ascends from left to right. If it's negative, the line descends. A zero slope indicates a horizontal line, and an undefined slope represents a vertical line.

• Positive slope: line rises
• Negative slope: line falls
• Zero slope: horizontal line
• Undefined slope: vertical line

Understanding slope is crucial for interpreting and predicting the behavior of linear equations.

Y-Intercept

The y-intercept is where a line crosses the y-axis of a graph. It is a fundamental part of the slope-intercept equation \( y = mx + b \). Here, \( b \) represents the y-intercept.

When graphing, the y-intercept provides a starting point. It's the initial value of \( y \) when \( x \) is zero. You simply find \( b \), plot it on the y-axis, and use it to start plotting the line.

• The y-intercept is the point (0, b)
• It is the value of \( y \) when \( x \) is zero
• Crucial for plotting the initial point of the line

Recognizing the y-intercept allows you to quickly see where the line starts on a graph.