Problem 37 - Understanding Elementary Algebra with Geometry

Question

Determine the slope of the line from its equation. $$y=5 x+7$$

Step-by-Step Solution

Answer

The slope is 5.

Step 1: Identify the Equation Form

Recognize that the given equation is in the slope-intercept form, which is written as: \( y = mx + b \), where \( m \) represents the slope and \( b \) represents the y-intercept.

Step 2: Compare to Standard Form

Compare the given equation \( y = 5x + 7 \) with the standard form \( y = mx + b \).

Step 3: Identify the Slope

From the comparison, observe that \( m = 5 \) in the given equation. Hence, the slope of the line is 5.

Key Concepts

Algebra

Algebra is a branch of mathematics that deals with symbols and the rules for manipulating those symbols. In algebra, these symbols (often letters) represent numbers in formulas and equations. By manipulating these symbols according to specific rules, we can solve equations and understand relationships between quantities.
In our example, we deal with an equation of the form:\[ y = 5x + 7 \].
Here, algebra helps us understand that this equation represents a straight line when graphed on a coordinate plane. The terms of the equation give us valuable information about the line's characteristics, such as its slope and y-intercept.
Understanding algebraic concepts like these is crucial because they are foundational for more advanced studies in mathematics and science.

Slope-Intercept Form

The slope-intercept form is a way of writing the equation of a straight line. It is one of the most common forms used because it clearly shows two important properties of the line: the slope and the y-intercept. The general formula for the slope-intercept form is: \[ y = mx + b \]
In this formula,

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

The slope tells us how steep the line is, and the y-intercept tells us the line's starting point on the y-axis.
In our given equation, \[ y = 5x + 7 \], it's clear that the equation follows the slope-intercept form with \( m = 5 \) and \( b = 7 \). This means the slope of the line is 5, and the line crosses the y-axis at the point (0, 7).

Slope Identification

Identifying the slope from an equation is straightforward when the equation is in slope-intercept form. The slope-intercept form is \( y = mx + b \), where the slope, \( m \), is the coefficient of the \( x \) term.
For the given equation, \[ y = 5x + 7 \], follow these steps to identify the slope:

• Recognize that the equation is already in the form \( y = mx + b \).
• Compare it to the standard form to find that \( m = 5 \).

Therefore, the slope of the line represented by the equation \( y = 5x + 7 \) is 5.
Understanding how to identify the slope is essential because the slope determines the direction and steepness of the line. A positive slope means the line rises as it moves from left to right, while a negative slope means it falls.
This skill is fundamental in algebra and helps in analyzing linear relationships in various contexts.