Problem 44 - Algebra A Combined Function

Question

Graph each linear equation. See Examples 4 through \(7 .\) \(9 x-6 y+3=0\)

Step-by-Step Solution

Answer

The linear equation forms a straight line with slope \(\frac{3}{2}\) and y-intercept \(\frac{1}{2}\). Plot it on a graph using these values.

Step 1: Rewrite the Equation in Slope-Intercept Form

First, we want to rearrange the given equation into the form \[y = mx + b\] where \(m\) is the slope and \(b\) is the y-intercept.Start with the original equation:\[9x - 6y + 3 = 0\]Subtract \(9x\) and \(3\) from both sides:\[-6y = -9x - 3\]Next, divide every term by \(-6\) to solve for \(y\):\[y = \frac{9}{6}x + \frac{3}{6}\]Simplify the fractions:\[y = \frac{3}{2}x + \frac{1}{2}\]

Step 2: Determine the Slope and Y-Intercept

From the equation \(y = \frac{3}{2}x + \frac{1}{2}\), we can identify the slope \(m\) and the y-intercept \(b\):- Slope \(m = \frac{3}{2}\)- Y-intercept \(b = \frac{1}{2}\)This tells us for every 2 units we move horizontally along the x-axis, the line moves 3 units vertically along the y-axis, starting at y-intercept \(\frac{1}{2}\).

Step 3: Plot the Y-Intercept

Graphically, start by plotting the y-intercept point on the graph:- The y-intercept \(\left(0, \frac{1}{2}\right)\) is where the line crosses the y-axis. Place a point at this location.

Step 4: Use the Slope to Find Another Point

From the y-intercept: - Use the slope \(\frac{3}{2}\), which means for every 2 units you go to the right (positive x-direction), you go up 3 units (positive y-direction).With the y-intercept of \(\left(0, \frac{1}{2}\right)\), move 2 units to the right to \(x = 2\) and 3 units up, reaching \(y = \frac{1}{2} + 3 = \frac{7}{2}\). Plot this point \((2, \frac{7}{2})\) on the graph.

Step 5: Draw the Line

Finally, draw a straight line through the two plotted points \((0, \frac{1}{2})\) and \((2, \frac{7}{2})\).This line represents the graph of the equation \(9x - 6y + 3 = 0\). Make sure the line extends across the graph to display its behavior as \(x\) approaches both positive and negative infinity.

Key Concepts

Graphing Linear Equations

When we graph linear equations, we aim to represent them visually on a coordinate plane. A linear equation results in a straight line. This is why these kinds of equations are referred to as "linear." Here’s a general idea of how this process works:

• Start by converting the equation into a form that makes it easy to identify its characteristics, like the slope and y-intercept.
• Find key points for accuracy on the graph. The y-intercept is often a good starting point because it provides a clear reference on the graph where the line intersects the y-axis.
• Use the slope (rise over run) to determine the direction and angle of the line.
• Drawing accurately requires connecting these points and extending the line on both sides.

By understanding these steps, you can confidently graph any linear equation and visualize its impact.

Slope-Intercept Form

The slope-intercept form of a linear equation is useful for graphing because it explicitly shows both the slope of the line and its y-intercept. Written as \(y = mx + b\), it provides straightforward insight:

• \(m\) represents the slope, determining how steep the line is and the direction it goes.
• \(b\) is the y-intercept, the point where the line crosses the y-axis.

One great advantage of converting an equation into slope-intercept form is the simplicity it gives you when plotting on a graph. With one point located (the y-intercept), the slope formula allows you to uncover the line's path by plotting additional points.

Slope and Y-Intercept

To understand a line fully, you need to grasp its slope and y-intercept. The slope, denoted as \(m\), determines the angle and direction of the line in the coordinate plane:

• A positive slope means the line ascends from left to right.
• A negative slope means the line descends as you move right.

The y-intercept, represented as \(b\), tells us where the line crosses the y-axis. This intercept gives us a starting point on the graph. Together, the slope and y-intercept allow you to reconstruct the line and understand its relationship within a system of coordinates.

Plotting Points

Plotting points is essential when graphing lines. You need at least two points to draw a line precisely:

• Begin with a known point, such as the y-intercept \((0, b)\).
• Use the slope to find another point. The slope \(m = \frac{rise}{run}\) means, from your initial point, move "rise" units up (or down if negative) and "run" units right.

For instance, from \((0, \frac{1}{2})\), if the slope is \(\frac{3}{2}\), you move 2 units right and 3 units up to plot a second point at \((2, \frac{7}{2})\). These two points can then be connected to form a straight line.

Equation Simplification

Simplifying an equation often involves rearranging it into a more convenient form for analysis or graphing, like the slope-intercept form. Start by isolating \(y\) on one side of the equation:

• Subtract or add terms to arrange \(y\) with other expressions on one side.
• Divide coefficients to get \(y\) alone.
• Simplify fractions or coefficients if necessary to make the equation clearer.

In our example, we restructured the equation \(9x - 6y + 3 = 0\) into \(y = \frac{3}{2}x + \frac{1}{2}\), making it straightforward to extract the slope and y-intercept. Simplification not only helps graphically but also strengthens the overall understanding of the equation's properties.