Problem 35 - Algebra 1

Question

Write the equation in slope-intercept form. Then graph the equation. $$ 3 x-6 y=9 $$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = \frac{1}{2}x - \frac{3}{2}\), and the line should cross the y-axis at -3/2 and have a slope of 1/2.

Step 1: Transpose the equation to slope-intercept form

The first step involves getting the equation into slope-intercept form (\(y = mx + b\)). To convert the equation \(3x - 6y = 9\) into slope-intercept form, isolate \(y\). First, divide all terms by 6 to get \(\frac{1}{2}x - y = \frac{3}{2}\). Then bring -\(y\) to the other side to get \(y = \frac{1}{2}x - \frac{3}{2}\).

Step 2: Identify the slope and y-intercept

Now that the equation is in slope-intercept form, it's easy to identify the slope, \(m\), and the y-intercept, \(b\). In the equation \(y = \frac{1}{2}x - \frac{3}{2}\), the slope \(m = \frac{1}{2}\) and the y-intercept \(b = -\frac{3}{2}\).

Step 3: Graph the equation

To graph the equation, start by having the y-intercept as the starting point in your drawing. In this case, the y-intercept is -3/2, so the line will cross the y-axis at -3/2. Then, use the slope to find the next point. The slope is 1/2, which can be understood as 'rise over run', meaning for each unit up (rise) along the y-axis, move 2 units right (run) along the x-axis. After these steps, draw a straight line that passes through these two points and this will be the graph of the equation!

Key Concepts

Graphing Equations

When it comes to graphing equations, especially in slope-intercept form, you're essentially drawing the visual path that an equation takes across a coordinate plane. This type of graphically showing an equation provides a clear picture of how the dependent and independent variables relate to each other.

To begin graphing any equation in slope-intercept form, identify two crucial components from the equation: the slope and the y-intercept. These two elements are like the instructions for your graph.

• Start by plotting the y-intercept on the y-axis. This point marks where the line will cross this vertical axis.
• Then utilize the slope to navigate and find the next point. The slope gives a directional guide, using 'rise over run', to determine the trajectory from one point to the next.
• With these two points plotted, simply connect them with a straight line and extend it across your coordinate plane. This represents the graph of your equation.

Graphing is a vital skill in visualizing mathematical relationships and can provide insights beyond numerical analysis.

Slope

The slope of a line is a measure of its steepness and direction as it travels across the plane. In mathematical terms, the slope is crucial because it quantifies how much a line inclines from the horizontal.

When working with linear equations like the slope-intercept form, the slope is denoted as \(m\) and appears as the coefficient of the \(x\)-term. Understanding the concept of slope involves recognizing:

• The slope value is a ratio known as 'rise over run', which means how much the line rises vertically for each unit it runs horizontally.
• If the slope is positive, the line ascends as it moves from left to right. Conversely, a negative slope means the line descends.
• A slope of zero indicates a perfectly horizontal line, while undefined or vertical lines do not have numerical slopes.

The slope is essentially the directional guideline for the graph, influencing how sharply or gently your line will traverse the plane.

Y-Intercept

The y-intercept plays an integral role in graphing linear equations. This point indicates where the line crosses the y-axis and provides a starting location for drawing the line.

In any equation structured in slope-intercept form, \(y = mx + b\), the y-intercept is represented by \(b\). It's a stand-alone term, unaffected by \(x\), and serves as a pivotal anchor for your line.

• The y-intercept corresponds to the value of \(y\) when \(x\) equals zero, essentially slicing through the y-axis.
• Getting this initial plot on the graph ensures that you have a fixed point to guide where your slope will take you next.
• By establishing the y-intercept first, you simplify the graphing process, making it easier to visualize and establish the line's path.

Understanding the y-intercept is crucial, not just for graphing, but in recognizing what happens in the equation when the independent variable vanishes.