Problem 28 - Brief Applied Calculus

Question

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

Step-by-Step Solution

Answer

Slope is \(\frac{2}{3}\) and \(y\)-intercept is \((0, -2)\).

Step 1: Simplify the equation

The given equation is \(y = \frac{2}{3}(x - 3)\). First, distribute the \(\frac{2}{3}\) to both \(x\) and \(-3\) inside the parentheses:\[ y = \frac{2}{3}x - \frac{2}{3} \times 3. \] This simplifies to \[ y = \frac{2}{3}x - 2. \] Now, the equation is in the slope-intercept form \(y = mx + b\).

Step 2: Identify the slope and y-intercept

In the equation \(y = \frac{2}{3}x - 2\), compare it to the standard slope-intercept form \(y = mx + b\). Here the slope \(m\) is \(\frac{2}{3}\) and the \(y\)-intercept \(b\) is \(-2\). The \(y\)-intercept as a point is \((0, -2)\).

Step 3: Plot the y-intercept

On a coordinate plane, mark the \(y\)-intercept \((0, -2)\). This is the point where the line crosses the \(y\)-axis.

Step 4: Use the slope to find another point

The slope \(\frac{2}{3}\) means rise over run. From the \(y\)-intercept \((0, -2)\), move up 2 units (rise) and 3 units to the right (run) to find another point, \((3, 0)\).

Step 5: Draw the line

Plot the points \((0, -2)\) and \((3, 0)\) on the graph. Connect these points with a straight line to represent the equation \(y = \frac{2}{3}x - 2\). The line represents all solutions to the equation.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that represent straight lines on a coordinate plane. They take the general form of \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept. This form is particularly useful as it directly gives us important information needed to graph these equations.

• The slope \( m \) indicates how steep the line is and in which direction it moves.
• The y-intercept \( b \) tells us where the line crosses the y-axis.

The simplicity of the slope-intercept form allows for quick identification of these components, making it easier to graph the equation and understand the relationship between variables.

Graphing

Graphing linear equations involves plotting points on a coordinate plane to visually represent the equation. To graph, we need:

• The slope \( m \).
• The y-intercept \( b \).

Start by plotting the y-intercept, as it is a fixed point where the line touches the y-axis. From this point, use the slope to determine the direction and steepness of the line. In our example, the slope \( \frac{2}{3} \) means for every 3 units you move horizontally, the line moves up 2 units.
Once another point is determined using the slope, draw a straight line through the points, extending across the coordinate plane. This line represents every possible solution to the linear equation.

Coordinate Plane

The coordinate plane is a two-dimensional space where we can graph and visualize equations. It consists of two perpendicular axes: the horizontal x-axis and the vertical y-axis.
Each point on the plane is represented as a pair \( (x, y) \), which indicates its position relative to these axes. For graphing linear equations, the y-intercept is where the line crosses the y-axis, and it's often our starting point on the graph.
Using the slope, we translate the line across the plane, plotting additional points to guide our line. This makes it easier to see solutions visually and understand how changes in the equation affect the line's position and orientation.