Problem 76 - Algebra 1: Concepts and Skills

Question

Find the slope and \(y\) -intercept of the graph of the equation. (Lesson 4.7) $$ y=-4 x+5 $$

Step-by-Step Solution

Answer

The slope of the graph of the given equation is -4 and the y-intercept is 5.

Step 1: Identify the Slope

The slope \(m\) of a linear equation is the coefficient of \(x\). In the provided equation, \(y = -4x + 5\), the slope \(m\) is -4.

Step 2: Identify the y-Intercept

The y-intercept \(b\) of a linear equation is the constant term. In the given equation, \(y = -4x + 5\), the y-intercept \(b\) is 5.

Key Concepts

Linear Equations

Understanding the basics of linear equations is crucial when exploring algebra and calculus. A linear equation is an algebraic equation in which each term is either a constant or the product of a constant and a single variable. Linear equations can be written in the form \(y = mx + b\), where \(m\) and \(b\) are constants, and \(x\) and \(y\) are variables representing coordinates on a graph.

The beauty of linear equations is that they graph as straight lines, hence the term 'linear.' The constant \(m\) represents the slope of the line, determining its steepness or incline, while \(b\), the y-intercept, tells us where the line crosses the y-axis. A solid grasp of this concept allows for analyzing and predicting relationships between variables, a common application across various scientific and economic fields.

Coefficient

In algebra, we often talk about the coefficient. But what exactly is a coefficient? It is a numerical or constant quantity placed before and multiplying the variable in an algebraic expression (it's the number in front of the variable). For example, in the equation \(y = -4x + 5\), \( -4\) is the coefficient of \(x\).

Importance of the Slope Coefficient

The coefficient of \(x\) in a linear equation is particular because it also represents the slope of the line. The slope tells us how quickly y changes for every unit change in x. If the coefficient (slope) is positive, the line will rise from left to right; if it's negative, it'll fall. A slope of zero means a horizontal line, indicating no change in y as x changes.

Constant Term

The constant term in a linear equation is the standalone number that doesn't change as the variables change. It is often represented as the \(b\) in the slope-intercept form \(y = mx + b\), and it provides unique insights into the graph of the equation.

For the equation \(y = -4x + 5\), the number \(5\) is our constant term. In graphical terms, this is the y-intercept, indicating that the line crosses the y-axis at the point (0, 5). Understanding the constant term is essential in graphing, as it serves as a starting point from which we can plot the rest of the line using the slope.

Graphing Linear Equations

The art of graphing linear equations involves translating algebraic equations onto a graph with an x and y-axis. This visual representation helps in understanding the relationship between variables. Let's take the earlier example \(y = -4x + 5\).

To graph this equation, we start by plotting the y-intercept (\(b\)) which is the point where the line will cross the y-axis. In our case, it is at the point (0, 5). Next, we apply the slope (\(m\)). Our slope is -4, which means for every unit increase in x, y decreases by 4 units. Starting from the y-intercept, we go one unit to the right (positive x direction) and four units down (negative y direction) to plot our next point. By connecting these points with a straight line, we have graphed our linear equation.