Problem 26 - 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 \(m = \frac{1}{2}\); y-intercept \((0, -2)\).

Step 1: Isolate y

First, we need to rearrange the equation to the standard form of a linear equation, which is \(y = mx + b\). Start by rewriting the original equation \(x = 2y + 4\). Subtract 4 from both sides to get \(x - 4 = 2y\). Then divide every term by 2 to isolate \(y\): \[y = \frac{x - 4}{2}\]This simplifies to:\[y = \frac{1}{2}x - 2\]

Step 2: Identify the Slope

Now that the equation is in the slope-intercept form \(y = mx + b\), identify the slope \(m\). In this form, the coefficient of \(x\) is the slope. Thus, the slope \(m = \frac{1}{2}\).

Step 3: Identify the y-intercept

The y-intercept is the constant term \(b\) in the equation \(y = mx + b\). Here, \(b = -2\), which means the y-intercept is the point \((0, -2)\).

Step 4: Graph the Line

With the slope and y-intercept in hand, plot the y-intercept \((0, -2)\) on the graph. Then, use the slope \(\frac{1}{2}\) to find another point. Starting from \((0, -2)\), move up 1 unit and to the right 2 units to plot the second point \((2, -1)\). Draw a straight line through these points to complete the graph.

Key Concepts

Slope-Intercept Form

One of the most important ways of expressing a linear equation is through the slope-intercept form. This form provides crucial information that makes graphing and understanding the equation much simpler. The slope-intercept form is expressed as:\[ y = mx + b \]

• \(m\) represents the slope of the line.
• \(b\) is the y-intercept, which is the point where the line crosses the y-axis.

This form is helpful because you can immediately determine both the slope and the y-intercept just by looking at the equation. This simplicity makes it easier to tackle problems involving graphing and solving linear equations. In the equation \(y = \frac{1}{2}x - 2\), the coefficient of \(x\) (\(\frac{1}{2}\)) gives the slope, while the constant term (\(-2\)) gives the y-intercept.

Graphing Lines

When graphing a line, starting with the y-intercept is often the easiest approach. This is because the y-intercept \((0, b)\) is a specific point on the graph where the line crosses the y-axis. In the equation we're considering, the y-intercept is \((0, -2)\).
Here’s how you can graph the line:

• First, plot the y-intercept \((0, -2)\) on the graph.
• Next, use the slope \(m\), which is \(\frac{1}{2}\) in our case. The slope tells you how to move from the y-intercept to another point on the line.

For a slope of \(\frac{1}{2}\), you move up 1 unit and to the right 2 units. This helps locate another point on the line, which would be \((2, -1)\). Once you have these two points, draw a straight line through them to extend your graph.

Algebraic Manipulation

Algebraic manipulation is the process of rearranging and simplifying equations. It's a key skill for solving problems involving linear equations, as it allows you to express equations in a more useful form, such as the slope-intercept form.
To isolate a variable, follow these steps:

• Start with the original equation \(x = 2y + 4\).
• Subtract 4 from both sides to get \(x - 4 = 2y\).
• Divide every term by 2: \(y = \frac{x - 4}{2}\).

Now, you have the equation in the form \(y = mx + b\). This form makes it easier to identify the slope and y-intercept, which are essential for graphing the line. With a few algebraic steps, you've transformed your equation into something that's immediately interpretable and actionable.