Problem 23 - Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. $$ 2 x-3 y=12 $$

Step-by-Step Solution

Answer

Slope: \(\frac{2}{3}\); Y-intercept: \(0, -4\).

Step 1: Rewrite the Equation in Slope-Intercept Form

We begin by rearranging the given equation to get it into the slope-intercept form, which is \(y = mx + b\). Starting with the given equation: \(2x - 3y = 12\). First, solve for \(y\) by isolating it on one side of the equation.Subtract \(2x\) from both sides:\[-3y = -2x + 12\]Next, divide each term by \(-3\):\[y = \frac{2}{3}x - 4\].This represents the equation in slope-intercept form.

Step 2: Identify the Slope and Y-intercept

In the equation \(y = \frac{2}{3}x - 4\), the slope \(m\) is the coefficient of \(x\), which is \(\frac{2}{3}\), and the \(y\)-intercept \((0, b)\) is \(-4\).

Step 3: Draw the Graph

To graph the equation, start by plotting the \(y\)-intercept point \((0, -4)\) on the graph. From this point, use the slope \(\frac{2}{3}\) to find another point on the line. This means moving up 2 units and right 3 units from \( (0, -4)\) to place another point at \((3, -2)\). Connect these points with a straight line to complete the graph.

Key Concepts

Slope Calculation

The slope of a line is a way to measure how steep the line is. It tells us how much the line rises or falls as we move from one point to another. To find the slope, we look at the coefficient of the variable \(x\) in the equation when it is written in the slope-intercept form \(y = mx + b\). In our example, once we transformed the equation to \(y = \frac{2}{3}x - 4\), we can see that \(m = \frac{2}{3}\).

• The slope \(m\) of \(\frac{2}{3}\) means that for every 3 units we move to the right on the graph, the line moves up by 2 units.
• A positive slope, like \(\frac{2}{3}\), indicates the line is increasing, meaning it goes upwards as we look from left to right.

Understanding the slope is essential for graphing and interpreting linear equations. It's about finding the pattern or direction of a line on a graph.

Y-intercept

The y-intercept is the point where the line crosses the y-axis on a graph. This occurs when the value of \(x\) is zero. In the equation written in the slope-intercept form \(y = mx + b\), the \(b\) represents the y-intercept.
For the equation \(y = \frac{2}{3}x - 4\), we identified that the y-intercept is \(-4\). This tells us that the line crosses the y-axis at point \((0, -4)\).

• The y-intercept gives us a starting point for graphing the line.
• It allows us to visualize part of the line without finding further points immediately.

By knowing the y-intercept, we have a clear depiction of where the line stands in relation to the origin on a Cartesian plane.

Graphing Linear Equations

Graphing linear equations is about transforming algebraic equations into visual lines on a coordinate plane. By understanding the slope and the y-intercept, we can quickly draw a line representing the equation. Here are the basic steps using the equation \(y = \frac{2}{3}x - 4\):

• First, identify and plot the y-intercept point \((0, -4)\) on the y-axis.
• Using the slope \(\frac{2}{3}\), move from the y-intercept point upward 2 units and to the right 3 units to find another point (\(3, -2\)).
• Mark this point on the graph.
• Draw a straight line through these points extending across the graph.

These steps help in effortlessly visualizing the equation. By following them, you get a clear and precise representation of how the equation behaves graphically. The line shows all the solutions to the equation, providing further insight into the relationship between \(x\) and \(y\) values in a real-world context.