Problem 36 - Algebra 1: Concepts and Skills

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 \(y = 0.5x - 1.5\). The graph is a line that crosses the y-axis at \(-1.5\) and rises 1 unit for every 2 units it runs to the right.

Step 1: Convert to Slope-Intercept Form

Start by isolating \(y\). Subtract \(3x\) from both sides of the equation to obtain: \(-6y = -3x + 9\). Then, divide every term by \(-6\) to isolate \(y\), resulting in: \(y = 0.5x - 1.5\)

Step 2: Identify the Slope and Y-intercept

The coefficient of \(x\) is the slope \(m\), and the constant term is the y-intercept \(b\). So, for the equation \(y = 0.5x - 1.5\), the slope \(m\) is \(0.5\) and the y-intercept \(b\) is \(-1.5\).

Step 3: Graph the Equation

Plot the y-intercept at \(-1.5\) on the y-axis. Since the slope is \(0.5\), rise by 1 unit and run by 2 units from the y-intercept point to get the next point. Continue this procedure to get more points. Finally, draw the line passing through these points.

Key Concepts

Solving Linear Equations

Solving linear equations is a fundamental skill in algebra that involves finding the value of a variable that makes the equation true. The objective is to isolate the variable, usually denoted as \(y\) or \(x\), on one side of the equation, making it easier to read and graph. Let's consider the equation \(3x - 6y = 9\). To solve for \(y\), we need to rearrange terms:

• First, move the term \(3x\) to the other side by subtracting it from both sides. This gives us \(-6y = -3x + 9\).
• Next, divide every term by \(-6\) to isolate \(y\). Doing so results in \(y = 0.5x - 1.5\).

By solving it in this way, we convert the equation into the slope-intercept form, making it straightforward to interpret and graph.

Graphing Linear Equations

Graphing linear equations allows us to visualize solutions. Once the equation is in slope-intercept form, like \(y = 0.5x - 1.5\), graphing becomes a step-by-step process:Begin by identifying the y-intercept and plot it on the y-axis. In our example, the y-intercept is \(-1.5\).
Then, use the slope to determine other points on the graph. The slope, represented as \(m\), is the rise over the run. For our equation, a slope of \(0.5\) means:

• For every 1 unit increase in \(y\) (vertical movement), move 2 units to the right on the \(x\) axis (horizontal movement).
• Repeat this pattern to plot several points.

Finally, connect these points with a straight line. Every point on the line is a solution to the equation, demonstrating the relationship between \(x\) and \(y\).

Slope and Y-Intercept

The slope and y-intercept are crucial in understanding the behavior of linear equations. The slope \(m\), in the equation \(y = mx + b\), describes the steepness and direction of the line:

• A positive slope, such as \(0.5\), indicates that as \(x\) increases, \(y\) also increases, and the line slants upwards to the right.
• If the slope were negative, the line would slant downwards.

The y-intercept \(b\) is where the line crosses the y-axis, which shows the value of \(y\) when \(x = 0\). For \(y = 0.5x - 1.5\), the y-intercept is \(-1.5\):

• This tells us that the line crosses the y-axis at \(-1.5\).
• Knowing the y-intercept helps quickly find a starting point on the graph.

Understanding these components helps in not only plotting the graph but also interpreting how changes in \(x\) affect \(y\). This makes predicting and solving linear systems much easier.