Problem 41 - Algebra and Trigonometry with Analytic Geometry

Question

Exer. 41-44: Use the slope-intercept form to find the slope and \(y\)-intercept of the given line, and sketch its graph. $$ 2 x=15-3 y $$

Step-by-Step Solution

Answer

Slope is \(-\frac{2}{3}\) and \(y\)-intercept is 5.

Step 1: Rearrange into Slope-Intercept Form

The slope-intercept form of a line is given by the equation \( y = mx + b \), where \( m \) is the slope and \( b \) is the \( y \)-intercept. To convert the given equation \( 2x = 15 - 3y \) into slope-intercept form, we first solve for \( y \) in terms of \( x \).

Step 2: Isolate the y-term

Start by moving the \( -3y \) term to the left side to get: \[ 3y = 15 - 2x \].

Step 3: Solve for y

Divide every term by 3 to isolate \( y \): \[ y = -\frac{2}{3}x + 5 \]. This is now in slope-intercept form \( y = mx + b \).

Step 4: Identify the Slope and y-Intercept

From the equation \( y = -\frac{2}{3}x + 5 \), the slope \( m \) is \( -\frac{2}{3} \), and the \( y \)-intercept \( b \) is 5.

Step 5: Graph the Line

To sketch the graph, start at the \( y \)-intercept (0,5) on the graph. From there, use the slope \( -\frac{2}{3} \) to determine the direction and steepness of the line. The slope indicates that for every 3 units you move to the right (positive direction on the x-axis), you move 2 units down (negative direction on the y-axis). Draw the line through the intercept using these points.

Key Concepts

Graphing Linear Equations

Graphing linear equations is an essential skill in algebra that visually represents relationships between variables. Linear equations are often expressed in the slope-intercept form, given by the equation \( y = mx + b \). The goal of graphing such an equation is to create a straight line on a coordinate plane that shows all possible solutions to the equation.

• Start by identifying the \( y \)-intercept \( b \). This point is where the line crosses the \( y \)-axis, which is at \( (0, b) \).
• From the \( y \)-intercept, use the slope \( m \) to find additional points.
• The slope \( m \) is a ratio, \( \frac{rise}{run} \), indicating how far up (or down) and how far across you move to find another point on the line.
• Plot these points meticulously and then draw a straight line through them to complete the graph.

Graphing helps in visualizing how the function changes across different \( x \) values, making it a powerful tool for understanding behaviors of functions quickly.

Finding Slope

Finding the slope of a line is a straightforward but crucial part of graphing and understanding linear equations. Slope is a measure of how steep a line is and can give insight into the relationship between two variables. In a linear equation in the form \( y = mx + b \), the slope is represented by \( m \).

• The slope \( m \) is calculated as \( \frac{rise}{run} \), or \( \frac{\Delta y}{\Delta x} \). This tells how much the \( y \) (vertical) value changes for a unit change in \( x \) (horizontal) value.
• If \( m \) is positive, the line rises from left to right, while a negative slope means it falls.
• A slope of zero signifies a horizontal line, indicating no change in \( y \) as \( x \) changes.
• Understanding the slope helps determine the steepness and direction of the line on a graph.

Being able to find and interpret the slope quickly allows you to predict how changes in one variable might affect another—a very practical skill in solving real-world problems.

Y-Intercept

The \( y \)-intercept is the point where a line crosses the \( y \)-axis on a graph. It gives an initial value of \( y \) when \( x \) is zero. In the slope-intercept form \( y = mx + b \), the \( y \)-intercept is represented by \( b \). Understanding and identifying the \( y \)-intercept is foundational in analyzing linear equations.

• The \( y \)-intercept \( b \) is straightforward to spot in the equation: it is the constant term.
• On a graph, the \( y \)-intercept provides a starting point for drawing the line.
• Knowing the \( y \)-intercept can also help in making predictions when the linear model represents real-world data.
• For example, if an equation models a business’s expenses \(( y )\) over time \(( x )\), the \( y \)-intercept indicates the fixed expenses when no time has passed.

Identifying the \( y \)-intercept accurately allows you to graph the line correctly and understand initial conditions in various scenarios.

Algebraic Manipulation

Algebraic manipulation involves rearranging equations to isolate desired variables or expressions. It is a powerful tool in solving equations, particularly when converting to the slope-intercept form. In our example, we started with the equation \( 2x = 15 - 3y \) and aimed to present it as \( y = mx + b \).

• First, bring all \( y \)-related terms to one side by adding or subtracting terms across the equation.
• Manipulate the equation by performing operations like multiplying or dividing each term to isolate \( y \).
• In our example, we rearranged to \( 3y = 15 - 2x \), and then divided by 3, resulting in \( y = -\frac{2}{3}x + 5 \).
• This manipulation revealed the slope \((-\frac{2}{3})\) and the \( y \)-intercept (5).

Mastering algebraic manipulation gives you the versatility to deal with various algebra problems, ensuring you can present equations in useful forms and interpret them correctly in various contexts.