Problem 13 - Introductory and Intermediate Algebra for College Students 4th

Question

In Exercises \(13-26,\) begin by solving the linear equation for \(y .\) This will put the equation in slope-intercept form. Then find the slope and the \(y\) -intercept of the line with this equation. $$-5 x+y=7$$

Step-by-Step Solution

Answer

The slope is 5 and the y-intercept is 7.

Step 1: Isolate y

First, the term with \(y\) needs to be isolated on one side of the equation. To achieve this, we add \(5x\) to both sides of the equation: \(-5x + y + 5x = 7 + 5x\), which simplifies to \(y = 5x + 7\).

Step 2: Identify the Slope and y-intercept

The slope-intercept form of a linear equation is given by \(y=mx+c\), where \(m\) is the slope and \(c\) is the y-intercept. Comparing this form with our equation \(y = 5x + 7\), it is clear that the slope \(m\) is 5 and the y-intercept \(c\) is 7.

Key Concepts

Linear Equations

Linear equations form the foundation of algebra and represent straight lines on a graph. They are called 'linear' because, no matter how long they are extended in either direction, they will never curve — their graph is always a straight line. A common form of a linear equation is:
\[ y = mx + b \]
where \( y \) is the dependent variable, \( x \) is the independent variable, \( m \) represents the slope of the line, and \( b \) denotes the y-intercept, which is the point where the line crosses the y-axis. A foundational skill in algebra is manipulating these equations to identify their slope and intercepts, thereby understanding their graphical behavior without needing to plot them.

Isolating Variables

Isolating variables is a critical process in algebra used to solve equations. The goal is to get the unknown variable on one side of the equation by itself, and everything else on the other side. This often involves performing the same operation on both sides of the equation to maintain equality. For example, if an equation has a term subtracted from the variable, you would add that term to both sides.

For instance: To isolate \( y \) in the equation \( -5x + y = 7 \), you would add \( 5x \) to both sides resulting in the isolated variable equation \( y = 5x + 7 \). Mastering the skill of isolating variables enables students to convert equations into different forms, such as slope-intercept form, and to solve for the unknowns within.

Slope and Y-Intercept

Understanding the slope and y-intercept of a linear equation is essential for graphing lines and analyzing their rate of change and initial value.
The slope (\( m \)): represents the steepness of the line, and is calculated as the rise over run between two points on the line. A positive slope means the line ascends from left to right, while a negative slope indicates it descends.
The y-intercept (\( b \)): is where the line crosses the y-axis. It indicates the value of \( y \) when \( x \) is zero. In the example \( y = 5x + 7 \), the slope is \( 5 \) and the y-intercept is \( 7 \).
This means for every unit increase in \( x \), the value of \( y \) increases by 5 units, and the graph of the equation crosses the y-axis at the point \( (0, 7) \).