Problem 42 - 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. $$\begin{aligned} &2 y-5 x=0\\\ &[-10,10] \text { by }[-10,10] \end{aligned}$$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \( y = \frac{5}{2}x \).

Step 1: Understand the Given Equation

The given equation is \( 2y - 5x = 0 \). This equation is in standard form \( Ax + By = C \), where \( A = -5 \), \( B = 2 \), and \( C = 0 \). Our task is to convert it to slope-intercept form \( y = mx + b \).

Step 2: Rearrange to Isolate 'y'

To convert the equation to slope-intercept form, we need to solve for \( y \). Start by moving the \( x \)-term to the other side of the equation: \( 2y = 5x \).

Step 3: Solve for 'y'

Divide every term by 2 to solve for \( y \):\[ y = \frac{5}{2}x\]Now, the equation is in the form \( y = mx + b \) where \( m = \frac{5}{2} \) and \( b = 0 \).

Step 4: Verify the Solution

Verify the solution by checking if putting it back in the original equation holds true. Substitute \( y = \frac{5}{2}x \) into the equation:\[2\left(\frac{5}{2}x\right) - 5x = 0 \5x - 5x = 0 \0 = 0\]The original and transformed equations are consistent, confirming the solution is correct.

Key Concepts

Standard Form

The standard form of a linear equation is represented as \( Ax + By = C \). Here, \( A \), \( B \), and \( C \) are integers or coefficients that make up the equation. In comparison to other forms, standard form is quite versatile. It allows us to easily identify both the intercepts on the axes by setting one of the variables to zero and solving for the other. Coaches and engineers often use this equation type to solve complex real-world problems.
When examining our original equation \( 2y - 5x = 0 \), it's clear this is in standard form. Here, \( A = -5 \), \( B = 2 \), and \( C = 0 \). Understanding this form is useful when transitioning to other forms, especially when graphing or interpreting straight line equations.

Equation Transformation

Transforming an equation involves rearranging its terms to convert it from one form to another. This process is essential because certain forms of an equation make it easier to understand different aspects of the equation's graphical representation.
In our case, converting a standard form equation to slope-intercept form \( y = mx + b \) lets us quickly identify the slope \( m \) and the y-intercept \( b \).

• Step 1: Start by isolating the term involving \( y \). Move the \( x \)-term to the other side: \( 2y = 5x \).
• Step 2: Solve for \( y \) by dividing all terms by \( 2 \): \( y = \frac{5}{2}x \).

This new form, \( y = \frac{5}{2}x \), reveals that the slope \( m \) is \( \frac{5}{2} \) and the y-intercept \( b \) is \( 0 \).

Graphing Equations

Graphing equations provides a visual representation of relationships within an equation. When a linear equation is in slope-intercept form, \( y = mx + b \), it becomes straightforward to graph the line. The graph shows the slope \( m \) directing how steep the line is, and the \( b \)-value indicates where the line crosses the y-axis.
For the equation \( y = \frac{5}{2}x \):

• The slope \( m = \frac{5}{2} \) suggests that for every 2 units the line moves horizontally, it ascends 5 units vertically.
• Since \( b = 0 \), the line crosses the origin point \((0, 0)\).

Using the given window range of \([-10,10] \text { by } [-10,10]\), it's beneficial to plot several points. Start at the origin, then apply the slope to mark additional points, ensuring they align on the same line for accuracy.

Linear Equations

Linear equations are integral in mathematics, describing relationships with constant rates of change. These equations produce straight lines on a graph, symbolizing proportional relationships between variables.
Key characteristics include:

• They involve no exponents higher than 1 on variables.
• Each term is either constant or the product of a constant and a single variable.

The concept behind linear equations extends to understanding real-world phenomena such as speed calculations, financial growth, and other cases showcasing steady changes. In this exercise, our linear equation \( y = \frac{5}{2}x \) represents a direct proportionality between \( x \) and \( y \), indicating as \( x \) increases or decreases, \( y \) does so at a predictable, steady rate.