Problem 38 - Elementary Algebra

Question

Determine the slope and \(y\) -intercept of the lines. $$ 2 y=9 x $$

Step-by-Step Solution

Answer

Answer: The slope of the line is 9/2 and the y-intercept is 0.

Step 1: Convert the given equation to the general form

To convert the given equation 2y = 9x to the general form y = mx + b, we need to solve for y. To do this, we divide both sides of the equation by 2: $$ y = \dfrac{9}{2}x $$

Step 2: Identify the slope and y-intercept

Comparing the converted equation to the general form y = mx + b, we can identify the slope (m) and y-intercept (b): $$ m = \dfrac{9}{2} $$ and $$ b = 0 $$

Step 3: Write down the slope and y-intercept

The slope of the line is: $$ m = \dfrac{9}{2} $$ and the y-intercept is: $$ b = 0 $$

Key Concepts

Linear Equations

Understanding linear equations is foundational in algebra. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can always be written in the form \( y = mx + b \) where:

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

This form of the equation is particularly useful because it gives a clear view of how the slope affects the steepness and direction of the line and how the y-intercept situates the line on the graph. Solving linear equations involves finding the values of 'y' for various values of 'x', or vice versa. They graph as straight lines, hence the term 'linear'—which means 'like a line'. The equation from the exercise, \( 2y = 9x \), is a linear equation. It can be rearranged to fit the format \( y = mx + b \( by dividing both sides by 2, resulting in the equation \( y = \frac{9}{2}x \( This equation shows a direct proportional relationship between 'y' and 'x', with the slope indicating how much 'y' changes for a unit change in 'x'.

Algebraic Slope

When studying linear equations, the concept of the algebraic slope is essential. The slope is a measure of the steepness, or incline, of a line, referred to by the variable 'm'. It's calculated as the ratio of the vertical change (rise) to the horizontal change (run) between two points on a line. So mathematically, it's expressed as:
\( m = \frac{\text{rise}}{\text{run}} \)
If you have two points, \( (x_1, y_1) \) and \( (x_2, y_2) \), the slope is \( m = \frac{y_2 - y_1}{x_2 - x_1} \).
A positive slope means the line ascends from left to right, while a negative slope means it descends. In the exercise's linear equation, \( 2y = 9x \), after transforming it to \( y = \frac{9}{2}x \), we can see that the slope is \( \frac{9}{2} \), indicating a relatively steep upward incline from left to right.

Y-intercept

The 'y-intercept' is another important concept when graphing and analyzing linear equations. It refers to the point where the line crosses the y-axis of a graph. The y-axis is a vertical line where the value of 'x' is zero, so the y-intercept is the value of 'y' when x equals zero. It's represented by the variable 'b' in the slope-intercept form of a line, \( y = mx + b \).

• The y-intercept gives one specific point on the graph: \( (0, b) \).
• It is the starting value of 'y' in many real-world interpretations when 'x' is nothing or the process just begins.

In our textbook exercise, once we converted the equation to \( y = \frac{9}{2}x \), it became clear that \( b = 0 \), meaning the line crosses the y-axis right at the origin (0,0). This tells us that there is no initial value added or subtracted from the line when 'x' is zero. Knowing the y-intercept is essential, as it facilitates the process of graphing the linear equation and understanding where the line will be positioned relative to the origin of the graph.