Problem 9 - Elementary Algebra

Question

For Problems 1-36, graph each linear equation. (Objective 2) $$ 3 x-y=6 $$

Step-by-Step Solution

Answer

Graph the line by plotting (0, -6) and (1, -3) and drawing a straight line through these points.

Step 1: Understand the Equation

The given equation is in the form \(3x - y = 6\). This is a linear equation representing a straight line on a graph.

Step 2: Rearrange to Slope-Intercept Form

The slope-intercept form is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept. First, solve for \(y\) in terms of \(x\) by adding \(y\) to both sides and then subtracting 6 from both sides: \(-y = -3x + 6\). Now, multiply each term by -1 to get \(y = 3x - 6\).

Step 3: Identify Slope and Y-intercept

In the equation \(y = 3x - 6\), the slope \(m\) is 3, and the y-intercept \(b\) is -6. This means the line crosses the y-axis at -6 and rises 3 units for every 1 unit it moves to the right.

Step 4: Plot the Y-intercept

Start by plotting the y-intercept on the graph. This point is (0, -6), where the line crosses the y-axis.

Step 5: Use the Slope to Find Another Point

From the point (0, -6), use the slope 3, which is the rise over run (3/1). Move up 3 units (rise) and 1 unit to the right (run) to find the next point at (1, -3).

Step 6: Draw the Line

With two points, (0, -6) and (1, -3), plot them on the graph. Then draw a straight line through these points, extending it across the graph to show the linear equation.

Key Concepts

Slope-Intercept Form

The slope-intercept form of a linear equation is an incredibly useful tool in graphing. It is written as \(y = mx + b\), where \(m\) and \(b\) are constants. Here, \(m\) represents the slope of the line, and \(b\) stands for the y-intercept, the point where the line crosses the y-axis.

The beauty of this form lies in its simplicity — you can quickly determine both the slope and the y-intercept… just by looking at the equation. This makes graphing lines a breeze, transforming a potentially complex process into one that feels much more manageable.

Always remember, converting any given equation into this form is your first step in graphing. By rearranging terms, you reposition the equation to easily reveal the line's characteristics.

Y-Intercept

The y-intercept is a vital aspect of understanding linear equations and their graphs. It's the point where the line meets the y-axis. On a graph, the y-intercept provides a fixed location that acts as a starting point for plotting the line.

In the slope-intercept form \(y = mx + b\), the constant term \(b\) represents the y-intercept. This is because when \(x = 0\), the equation simplifies to \(y = b\), making \((0, b)\) the coordinates of the y-intercept.

When plotting, you first locate this intercept on the y-axis. It acts like an anchor, giving you a perspective on how the line stretches across the graph. For example, with the equation \(y = 3x - 6\), the y-intercept is -6, meaning the line crosses the y-axis at this point.

Graphing Linear Equations

Graphing linear equations is a fundamental skill in algebra, connecting the abstract world of equations with the visual world of graphs. The process involves plotting points and drawing a line through these points to represent the equation.

To start graphing, convert the equation to slope-intercept form if necessary. Then plot the y-intercept, which gives your line a home base. For example, in \(y = 3x - 6\), the y-intercept is \(-6\), so plot the point (0, -6).

Use the slope, which is rise over run, to move from the y-intercept to another point. In this case, the slope is 3, meaning from (0, -6), move up 3 units and 1 unit to the right to plot the point (1, -3).

• Start with y-intercept
• Use slope to find next point
• Connect points with a line

Drawing a line through these points completes your graph, offering a straightforward visual representation of the equation.