Problem 34 - Elementary Algebra

Question

For the following problems, determine the slope and \(y\) -intercept of the lines. $$ -5 y=15 x+55 $$

Step-by-Step Solution

Answer

Answer: The slope of the line is -3 and the y-intercept is -11.

Step 1: Rewrite the equation in slope-intercept form

To rewrite the equation in slope-intercept form, we'll start by dividing both sides of the equation by -5. This will help to isolate y on the left side of the equation. $$ \frac{-5 y}{-5}=\frac{15 x + 55}{-5} $$ Now simplify the equation: $$ y= -3x -11 $$

Step 2: Identify the slope and y-intercept

In the slope-intercept form, \(y = mx + b\), 'm' represents the slope and 'b' represents the y-intercept. Comparing this with our equation \(y = -3x - 11\), we can identify: $$ m = -3 \quad \text{and} \quad b = -11 $$ So, the slope of the line is -3 and the y-intercept is -11.

Key Concepts

Understanding the Slope

When we talk about the slope of a line in a linear equation, we're referring to a number that describes the steepness or incline of the line. In the slope-intercept form of a linear equation, which is written as \( y = mx + b \), the letter \( m \) represents the slope. This value tells us how much \( y \) changes for a unit change in \( x \). For instance,

• If the slope is positive, the line rises as it moves from left to right.
• If the slope is negative, the line falls as it moves from left to right.
• If the slope is zero, the line is perfectly horizontal.

In the given equation \( y = -3x - 11 \), the slope \( m \) is -3. This means for every 1 unit increase in \( x \), \( y \) decreases by 3 units. The negative sign indicates that the line slopes downward.

Identifying the Y-Intercept

The y-intercept in a linear equation is the point where the line crosses the y-axis. In slope-intercept form \( y = mx + b \), the \( b \) represents the y-intercept. This value is crucial because it shows the initial position of the line on a graph when \( x = 0 \). To find the y-intercept:

• Set \( x \) to zero in the equation.
• Solve for \( y \).

For example, in the equation \( y = -3x - 11 \), replacing \( x \) with 0 gives us \( y = -11 \). Thus, the y-intercept is \( -11 \). This means the line intersects the y-axis at the point (0, -11). It's a helpful way to graph the line quickly.

Basics of Linear Equations

Linear equations represent straight lines on a graph and are fundamental in algebra. The slope-intercept form, \( y = mx + b \), is a simple and effective way to express these equations:

• It quickly provides both the slope and y-intercept, helping in graph plotting.
• This form is highly intuitive for evaluating how changes in \( x \) affect \( y \).

Linear equations model everyday scenarios where there is a constant rate of change. For instance:

• Calculating distance over time if speed is constant.
• Determining cost per unit in budgeting scenarios.

By understanding these equations, you can solve practical problems and better understand mathematical relationships.