Problem 56 - Algebra 1

Question

Find the slope and the \(y\) -intercept of the graph of the equation. Then graph the equation. $$ x+6 y=12 $$

Step-by-Step Solution

Answer

The slope of the given equation is \(-\frac{1}{6}\), and the y-intercept is \(2\). The graph of the equation is a line that passes through the y-intercept and slopes downwards.

Step 1: Rewrite the equation in \(y = mx + b\) form

To do this, we will start with the given equation \(x+6y=12\) and rearrange it: subtract \(x\) from both sides to isolate \(6y\), then divide every term by \(6\). The result will be \(y = -\frac{1}{6}x + 2\).

Step 2: Identify the slope and the y-intercept

From the equation in step 1, we can identify the slope and y-intercept. The slope \(m\) is the coefficient of \(x\), which is \(-\frac{1}{6}\), and the y-intercept \(b\) is the constant term, which is \(2\). So, our slope is \(-\frac{1}{6}\) and y-intercept is \(2\).

Step 3: Graph the equation

Firstly, we mark the y-intercept at \(2\) on the y-axis. The negative slope indicates that the graph will slope downwards; since our slope is \(-\frac{1}{6}\), we will go down 1 unit and right 6 units from the y-intercept to plot the next point. Then we draw the line that passes through these points, which is the graph of the equation.

Key Concepts

Linear Equations

Linear equations are fundamental in algebra, representing a straight line when graphed. These equations are generally in the form \(y = mx + b\), known as the slope-intercept form. This form makes it simple to identify crucial characteristics such as the slope and the y-intercept of the equation. In the equation \(x + 6y = 12\), we can rearrange it to reveal its slope-intercept form by isolating \(y\) on one side. This rearrangement helps in understanding how the equation represents a line graphically. Linear equations can have different forms but transforming them into the slope-intercept form simplifies the process of graphing and analyzing them. Remember, linear equations are defined by constants and single variables, maintaining a degree of 1 for the variable.

Graphing

Graphing a linear equation provides a visual representation of all solutions to that equation. To graph the equation \(y = -\frac{1}{6}x + 2\), we begin by plotting the y-intercept, which is where the line crosses the y-axis. By identifying and plotting additional points using the slope, a line is constructed. This process allows us to see the continuous nature of a linear relationship.
Once you understand how to plot these components, any linear equation can be easily graphed by hand or with graphing tools. The visual display also helps in interpreting the direction and steepness of the line, indicated by the slope.

Slope

The slope of a line in a linear equation is denoted by the letter \(m\) in the slope-intercept form \(y = mx + b\). It measures how much the line rises or falls as you move along the x-axis. For the equation \(y = -\frac{1}{6}x + 2\), the slope is \(-\frac{1}{6}\).

• A positive slope means the line is rising.
• A negative slope indicates the line is falling.

In our equation, the negative slope tells us the line will descend as it moves from left to right. The smaller or larger the absolute value of the slope, the more gentle or steep the line will be. The slope is crucial for plotting the direction of the line accurately on a graph.

Y-Intercept

The y-intercept is an important feature where the line crosses the y-axis, denoted by \(b\) in the slope-intercept form. In the equation \(y = -\frac{1}{6}x + 2\), the y-intercept is \(2\). This point is significant because it provides a starting point for plotting the graph.
The y-intercept is easy to find since it is the constant term in the equation. To plot your graph, begin at this point on the y-axis. From here, use the slope to find other points along the line. The y-intercept is very useful in real-world applications, marking the initial condition or output when zero inputs are plugged into the equation, such as a starting balance in economic models or a baseline measurement in experiments.