Problem 61 - Algebra 2

Question

State the slope and the y-intercept of the graph of each equation. \(2 y=6 x+14\)

Step-by-Step Solution

Answer

Slope: 3, Y-intercept: 7

Step 1: Convert to Slope-Intercept Form

The equation given is \(2y = 6x + 14\). To find the slope and y-intercept, we need to express the equation in slope-intercept form, which is \(y = mx + b\), where \(m\) is the slope and \(b\) is the y-intercept.First, divide every term by 2 to solve for \(y\):\[ y = \frac{6x}{2} + \frac{14}{2} \]Simplifying gives:\[ y = 3x + 7 \]

Step 2: Identify the Slope

In the equation \(y = 3x + 7\), the coefficient of \(x\) is the slope. Here, the slope \(m\) is 3.

Step 3: Identify the Y-Intercept

In the equation \(y = 3x + 7\), the constant term is the y-intercept. Here, the y-intercept \(b\) is 7.

Key Concepts

Understanding Linear Equations

A linear equation is a fundamental concept in algebra. It represents a straight line when graphed on a coordinate plane. The general form of a linear equation is often written as \(y = mx + b\), known as the slope-intercept form.
This form makes it easy to understand the characteristics of the line, such as its slope and y-intercept. The variable \(x\) is the independent variable, while \(y\) is the dependent variable affected by \(x\). Linear equations are used to model relationships between two variables and predict how changes in one affect the other.
They are simple, yet powerful tools that help describe relationships in various real-world scenarios, such as calculating speed, budgeting, and even scientific phenomena. Moving forward, it is crucial to become familiar with linear equations because they are foundational for understanding more complex algebraic concepts.

Slope Calculation and Its Significance

One of the key features of any line described by a linear equation is its slope. The slope is represented by the letter \(m\) in the slope-intercept form \(y = mx + b\). It measures the steepness of the line and indicates the direction.
To calculate the slope, you simply look at the coefficient of \(x\) in the slope-intercept form of the equation. For the equation \(y = 3x + 7\), the slope \(m\) is 3. This means that for every unit increase in \(x\), \(y\) increases by 3 units.

• A positive slope, like 3, means the line rises as you move from left to right.
• A negative slope would mean the line falls as you move from left to right.
• A zero slope indicates a horizontal line, where \(y\) remains constant despite changes in \(x\).

Understanding the slope is crucial because it determines how changes in one variable affect the other. It's an essential concept in analyzing linear relationships and trends.

Calculating the Y-Intercept

The y-intercept of a line is the point where the line crosses the y-axis. In the slope-intercept form of a linear equation, \(y = mx + b\), the y-intercept is denoted by \(b\).
For example, in the equation \(y = 3x + 7\), the y-intercept \(b\) is 7. This means that when \(x = 0\), \(y\) is equal to 7. Visually, it's the point at which the line intersects the y-axis on a graph.

• The y-intercept represents the starting value of \(y\) when \(x\) is zero.
• It provides a basis for understanding linear relationships without the influence of other variables.
• Knowing the y-intercept can help determine how a line behaves near the y-axis.

Calculating the y-intercept is straightforward when you have an equation in slope-intercept form, but it's vital to interpret it correctly within the context of real-world applications and scenarios.