Problem 38 - A Graphical Approach to Precalculus with Limits

Question

Write each equation in the form \(y=m x+b .\) (A suggested window for a comprehensive graph of the equation is given. $$\begin{aligned} &6 x+5 y=9\\\ &[-10,10] \text { by }[-10,10] \end{aligned}$$

Step-by-Step Solution

Answer

The equation in the form \(y=mx+b\) is \(y = -\frac{6}{5}x + \frac{9}{5}\).

Step 1: Start with the original equation

We begin with the given equation: \(6x + 5y = 9\). Our goal is to rearrange this equation into the form \(y = mx + b\).

Step 2: Isolate the y-term

Subtract \(6x\) from both sides to move the \(x\)-term to the other side:\[5y = -6x + 9\]

Step 3: Solve for y

Divide every term by 5 to solve for \(y\):\[y = \frac{-6}{5}x + \frac{9}{5}\]

Step 4: Final form of the equation

The equation \(y = mx + b\) form is reached, where \(m = -\frac{6}{5}\) and \(b = \frac{9}{5}\). This is the slope-intercept form of the line. \(y = -\frac{6}{5}x + \frac{9}{5}\).

Key Concepts

Linear Equations

Linear equations are foundational in algebra and are used widely in mathematics to express relationships between variables. A linear equation is an equation between two variables that gives a straight line when plotted on a graph. The most common form of a linear equation is the slope-intercept form, which is written as:\[ y = mx + b \]In this formula:

• \(y\) is the dependent variable, usually represented on the vertical axis.
• \(x\) is the independent variable, usually represented on the horizontal axis.
• \(m\) is the slope of the line, indicating the steepness and direction.
• \(b\) is the y-intercept, or where the line crosses the vertical axis.

Linear equations can represent many real-world situations, such as calculating distances over time, pricing items over quantities, and more. Understanding how to manipulate and interpret these equations is key to solving many mathematical and practical problems. That’s where techniques like rearranging equations come in handy.

Rearranging Equations

Rearranging equations involves modifying the structure of an equation to make it easier to analyze or solve. For our exercise, the goal is to rearrange a given equation into the slope-intercept form, which helps to find the slope and y-intercept easily. Let’s see how this is done:

Suppose we start with the equation:\[ 6x + 5y = 9 \]Our objective is to isolate \(y\) on one side of the equation. Here's how to do this step-by-step:

• Step 1: Subtract \(6x\) from both sides to move all terms involving \(x\) to the other side: \[ 5y = -6x + 9 \]
• Step 2: Divide every term by \(5\) to solve for \(y\): \[ y = \frac{-6}{5}x + \frac{9}{5} \]

Now, the equation is in slope-intercept form. Rearranging helps to clarify the line's characteristics, preparing it for graphing or further analysis.

Graphing Linear Equations

Graphing linear equations involves plotting points on a coordinate plane to display the straight line represented by an equation. The slope-intercept form \(y = mx + b\) simplifies this task by directly revealing the slope and y-intercept.

For example, take the equation:\[ y = -\frac{6}{5}x + \frac{9}{5} \]Here's how to graph this equation step-by-step:

• Step 1: Identify the y-intercept, \(b\). This is \(\frac{9}{5}\), so you will start by marking this point on the y-axis.
• Step 2: Use the slope \(m = -\frac{6}{5}\), which tells you the rise and run. From the y-intercept, you move down 6 units and right 5 units to find the next point.
• Step 3: Connect these points with a straight line. This line represents all solutions to the equation.

Graphing these equations allows one to visually interpret the relationship between variables, see trends, and predict values, making it a powerful tool in both educational and real-world settings.