Problem 30 - Algebra 1: Concepts and Skills

Question

Find the \(y\) -intercept of the line. $$ y=6 x-24 $$

Step-by-Step Solution

Answer

The y-intercept of the line given by the equation \(y=6x-24\) is -24.

Step 1: Identify the Slope-Intercept Form

Firstly, recognize that the equation \(y = 6x - 24\) is in slope-intercept form, that is \(y = mx + b\). Here, m is the slope, and b is the y-intercept.

Step 2: Identify the y-Intercept

In the given equation \(y = 6x - 24\), compare it with the standard form to identify the y-intercept. It is clear from the comparison that the number following the x term, which is -24, is the y-intercept.

Key Concepts

Slope-Intercept Form

The slope-intercept form is a straightforward way to express the equation of a straight line. It is written as \(y = mx + b\), where:\- \(m\) represents the slope of the line. The slope indicates how steep the line is and the direction it is going. You can think of it as the rise over run, or how much \(y\) changes for a unit change in \(x\). \- \(b\) indicates the y-intercept, the point where the line crosses the y-axis.One important aspect of the slope-intercept form is its ability to make graphing a line a breeze. With just the slope and the y-intercept, you can quickly sketch the linear equation. Recognizing this form makes it easy to find these essential characteristics of a line.

Linear Equations

Linear equations are equations of the first degree, meaning the variable(s) is not raised to any power higher than one. As opposed to quadratic or cubic equations, linear equations represent straight lines when plotted on a graph.A linear equation can be presented in various formats, but the slope-intercept form \(y = mx + b\) is one of the most common. This form is particularly useful because it provides direct insight into the slope and y-intercept.Generally, linear equations:

• Only have terms that are either constants or products of a constant and a single variable.
• Show a constant rate of change between the variables.
• Are fundamental in understanding more complex mathematical concepts.

They are commonly seen in algebraic contexts and are essential for beginners who are getting to grips with the basics of algebra.

Algebraic Expressions

Algebraic expressions are mathematical phrases that can involve numbers, variables, and operations. They do not have an equality sign, unlike equations. However, they are a crucial building block for forming equations.In expressions like \(6x - 24\), found in the slope-intercept form \(y = mx + b\), 6x represents a term where 6 is the coefficient of the variable \(x\). The \(x\) is the variable that can change its value. The constant term here, -24, acts independently of \(x\) and shifts the graph up or down.Working with algebraic expressions involves:

• Simplifying them by combining like terms.
• Evaluating them by substituting the values for the variables.
• Understanding how they form the backbone of more complex algebraic tasks.

Grasping algebraic expressions is vital for progressing to solving linear equations and understanding various algebra concepts.