Problem 108 - Introductory and Intermediate Algebra for College Students 4th

Question

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. The graph of any equation in the form \(y=m x+b\) passes through the point \((0, b)\)

Step-by-Step Solution

Answer

The statement 'The graph of any equation in the form \(y=m x+b\) passes through the point (0, b)' is True.

Step 1: Understanding the linear equation form

The equation given is in the slope-intercept form of a linear equation, which is \(y = mx + b\). In this equation, 'm' represents the slope of the line and 'b' is the y-intercept, which is the point at which the line crosses the y-axis.

Step 2: Insert point into equation

To verify the statement, we should substitute the point (0, b) into the equation. This point means that for \(x=0\), \(y=b\).

Step 3: Check whether the point satisfies the equation

Inserting the values we get \(b = m*0 + b\), which simplifies to \(b = b\). It is evident that the given point is the solution of the equation.

Step 4: Final statement

Because the equation \(y = m*x + b\) holds true for the point (0, b), it can be stated that the graph of any equation in the form \(y=m x+b\) passes through the point (0, b). The statement in the question is True.

Key Concepts

Linear Equations

The foundation of algebra lies in understanding linear equations, which are formulas that represent straight lines when graphed on a Cartesian plane. A linear equation can be identified by its standard form, which is \(Ax + By = C\)), or more commonly by its slope-intercept form \(y = mx + b\)). In these expressions, \(A\)), \(B\)), and \(C\)) are constants that determine the specific attributes of the line, whereas \(m\)) denotes the slope and \(b\)) represents the y-intercept.

Concisely, the slope \(m\)) indicates how steep the line is, whereas the y-intercept \(b\)) is the point where the line crosses the y-axis. Recognizing the slope-intercept form is crucial because it provides direct insights into these two characteristics of the line with minimal effort. It's the most efficient form for graphing purposes and helps to easily predict the behavior of linear relationships commonly found in various real-world applications.

Y-intercept

The y-intercept is a fundamental concept in the study of linear equations. It's the point where the line crosses the y-axis and is often represented as (0, b) in the coordinate plane. This b value is a key component of the slope-intercept form \(y = mx + b\)). The y-intercept gives us critical information about the line: no matter what the slope may be, if you look at the graph where x equals zero, you will find the y-intercept.

Understanding the y-intercept is vital for graphing and interpreting linear equations because it serves as a starting point for plotting the line. When the equation of a line is given, students should be adept at quickly identifying the y-intercept, which immediately positions the line in reference to the y-axis and sets the stage for further analysis or comparison with other lines.

Graphing Linear Equations

Graphing linear equations involves translating the information from an algebraic equation onto a visual graph. Using the slope-intercept form \(y = mx + b\)), we start by plotting the y-intercept (0, b) on the graph. This is where the line will intersect the y-axis. Next, we use the slope \(m\)), which is the ratio of the rise over run, to determine the direction and steepness of the line, guiding us to subsequent points.

For a positive slope, the line rises to the right, and for a negative slope, it falls to the right. If the slope is zero, the line is horizontal, indicating a constant y-value across all x-values. With the correct plot of the y-intercept and application of the slope, the line can then be drawn extending infinitely in both directions, which represents all the solutions to the linear equation. This visual representation aids in understanding various concepts such as intersection points with other lines, which can represent solutions to systems of equations.