Problem 21 - 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. $$2 x+7 y=0$$

Step-by-Step Solution

Answer

The equation in slope-intercept form is \(y = -\frac{2}{7}x\). The slope of the line is \(-\frac{2}{7}\) and the y-intercept is \(0\).

Step 1: Rearrange the Equation

The first thing to do is to solve the equation \(2x + 7y = 0\) for \(y\). To do this, move \(2x\) to the right side of the equation by subtracting \(2x\) from both sides. The equation becomes \(7y = -2x\).

Step 2: Solve for y

We now need \(y\) to be by itself, so divide every term in the equation by \(7\). Doing so gives us the equation \(y = -\frac{2}{7}x\).

Step 3: Identify the Slope and Y-Intercept

This equation is now in the slope-intercept form \(y = mx + b\). So, we see that the slope \(m\) is \(-\frac{2}{7}\) and the y-intercept \(b\) is \(0\) (since there's no constant added or subtracted on the end).

Key Concepts

Slope-Intercept Form

Linear equations are often expressed in different forms, but one of the most popular and simplest is the slope-intercept form. The primary structure of this form is expressed as \(y = mx + b\). Here, \(m\) represents the slope of the line, and \(b\) is the y-intercept. This format makes it exceptionally easy to graph a line on a coordinate plane.

In our given example, we rearranged the equation \(2x + 7y = 0\) to get \(y = -\frac{2}{7}x\). This is a perfect representation of the slope-intercept form. Notice that there's no additional number at the end of this equation, meaning that the y-intercept \(b\) is \(0\), indicating that the line crosses the y-axis at the origin.

Solving Equations

To solve the given linear equation for \(y\), we start by rearranging all terms to isolate it. We initially had the equation \(2x + 7y = 0\). To isolate \(y\), we subtract \(2x\) from both sides. This action transforms the equation into \(7y = -2x\).

This equation still has \(y\) multiplied by \(7\). To finally solve for \(y\), we divide both sides by \(7\). This results in \(y = -\frac{2}{7}x\). Through these steps, it's crucial to perform the same operation on both sides of the equation to maintain equality. Maintaining the balance ensures the solution stays accurate and meaningful.

Algebra Basics

Understanding the foundation of algebra includes knowing how to manipulate and solve equations. Linear equations with variables typically involve basic operations like addition, subtraction, multiplication, and division.

In our example, we extracted these basics to rearrange the equation \(2x + 7y = 0\) into the form \(y = -\frac{2}{7}x\). We used subtraction to move the \(2x\) term and then division to isolate \(y\). Simply put, solving an equation often requires simplifying expressions and moving terms across the equal sign. We always aim to isolate the variable of interest, which often means simplifying the equation to a form that is easy to read and interpret.

• Think of solving equations as a method of organization: line up like terms and simplify whenever possible.
• Use inverse operations to undo any operations and isolate the variable.
• Check your final expression to ensure it's in the desired form, such as the slope-intercept form used above.