Problem 26 - Brief Applied Calculus

Question

For each equation, find the slope \(m\) and \(y\) -intercept \((0, b)\) (when they exist) and draw the graph. \(x=2 y+4\)

Step-by-Step Solution

Answer

Slope is \(\frac{1}{2}\), \(y\)-intercept is \((0, -2)\).

Step 1: Rewrite the Equation in Slope-Intercept Form

Start by rewriting the equation \(x = 2y + 4\) to get it into the slope-intercept form \(y = mx + b\). The slope-intercept form makes it easier to identify the slope \(m\) and the \(y\)-intercept \((0, b)\).First, solve for \(y\) by isolating it:\[x = 2y + 4\]Subtract 4 from both sides:\[x - 4 = 2y\]Divide each term by 2 to solve for \(y\):\[y = \frac{x}{2} - 2\]Now, the equation is in slope-intercept form \(y = \frac{1}{2}x - 2\).

Step 2: Identify the Slope and Y-Intercept

From the equation \(y = \frac{1}{2}x - 2\), identify the slope \(m\) as the coefficient of \(x\), which is \(\frac{1}{2}\), and the \(y\)-intercept \((0, b)\) as \(-2\). This tells us:- Slope (\(m\)) is \(\frac{1}{2}\).- \(y\)-intercept is \((0, -2)\).

Step 3: Draw the Graph

To draw the graph, start by plotting the \(y\)-intercept \((0, -2)\) on the coordinate plane. Then, use the slope \(\frac{1}{2}\) to plot another point. From the \(y\)-intercept, move up 1 unit and right 2 units, since the slope \(\frac{1}{2}\) means 'rise over run'. This gives you another point: \((2, -1)\).Draw a line through the two points \((0, -2)\) and \((2, -1)\) to complete the graph.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that form a straight line when graphed on a coordinate plane. Each linear equation can be written in different forms, but the most commonly used form is the slope-intercept form, which is: \[ y = mx + b \]Here:

• \( m \) is the slope of the line. It describes how steep the line is. A positive slope means the line ascends, while a negative slope indicates a descending line.
• \( b \) is the y-intercept. This is the point where the line crosses the y-axis.

To solve a given equation in order to put it in slope-intercept form, you typically solve for \( y \), allowing you to identify its slope and y-intercept easily. For example:- Starting from the equation \( x = 2y + 4 \), we need to isolate \( y \) to express it in the slope-intercept form.- Through rearranging, you get \( y = \frac{1}{2}x - 2 \) meaning the slope \( m \) is \( \frac{1}{2} \), and the y-intercept \( b \) is \(-2\). This formula allows you to quickly understand the direction and position of the line based on the values of slope \( m \) and y-intercept \( b \).

Graphing

Graphing is a crucial skill in understanding linear equations visually. Once you have a linear equation in the slope-intercept form, you can easily graph it by following these steps:

• Begin by identifying the y-intercept \( b \). This tells you where your line will cross the y-axis. For example, if \( b = -2 \), you place your first point at \((0, -2)\).
• Use the slope \( m \) to plot another point starting from the y-intercept. The slope \( \frac{1}{2} \) suggests a rise of 1 unit for every 2 units run, or right. From the point \((0, -2)\), move up 1 unit and over 2 units to the right, placing the next point at \((2, -1)\).
• To complete the graph, draw a straight line through these points. Extend the line across the grid as desired. This line represents all solutions to the equation \( y = \frac{1}{2}x - 2 \).

Graphing helps to visualize the relationship between \( x \) and \( y \) in an equation and provides a clear picture of how changes in these variables affect the line's position and slope.

Coordinate Plane

The coordinate plane is a two-dimensional surface on which you can graph points, lines, and curves. It consists of a horizontal number line known as the x-axis, and a vertical number line, called the y-axis. These axes intersect at a point called the origin, denoted as \((0, 0)\). When graphing a linear equation, like in our example, this serves as the stage for plotting points and drawing lines:

• Each point on the plane represents a pair of values from the equation \((x, y)\).
• First, locate the y-intercept \((0, b)\), where the line crosses the y-axis.
• Next, use the slope to determine the second point. You can then draw the line to show the full equation.

Coordinate planes are fundamental not just in learning algebra but also in many fields of science and engineering where graphing relationships visually is essential. Understanding how to navigate the coordinate plane is core to succeeding in graphing linear equations successfully.