Problem 43 - Understanding Elementary Algebra with Geometry

Question

Determine the slope of the line from its equation. $$2 y+3 x=6$$

Step-by-Step Solution

Answer

The slope is \(-\frac{3}{2}\).

Step 1 - Rewrite the equation in slope-intercept form

The slope-intercept form of a line equation is given by \( y = mx + b \). To rewrite the given equation \(2y + 3x = 6\) in this form, first subtract \(3x\) from both sides to isolate the \(y\) term on one side:\[ 2y = -3x + 6 \]

Step 2 - Solve for \( y \)

Next, divide every term in the equation \(2y = -3x + 6\) by 2 to completely isolate \(y\):\[ y = -\frac{3}{2}x + 3 \]

Step 3 - Identify the slope

In the slope-intercept form \( y = mx + b \), the coefficient of \(x\) represents the slope \(m\). From the equation \( y = -\frac{3}{2}x + 3 \), the slope \(m\) is \(-\frac{3}{2}\).

Key Concepts

slope-intercept form

The slope-intercept form is a convenient way to express linear equations. The general format is \(y = mx + b\), where \(m\) is the slope of the line, and \(b\) is the y-intercept (the point where the line crosses the y-axis).
To make an equation easier to work with, especially for finding the slope, you should convert it to the slope-intercept form.
For example, if you have the equation \(2y + 3x = 6\), you can rearrange it into the slope-intercept form by isolating \(y\). This will make it simpler to identify both the slope \(m\) and the y-intercept \(b\).

linear equations

Linear equations represent lines on a graph. These equations can be written in various forms, but the slope-intercept form is one of the most commonly used formats.
In general, a linear equation can be written as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are constants.
For instance, consider the equation \(2y + 3x = 6\). This is a standard form of a linear equation.
To convert it to the slope-intercept form, you need to manipulate it so that it looks like \(y = mx + b\).
Recognizing how to transform different forms of linear equations is crucial for solving problems related to lines on a graph.

isolating variable

Isolating the variable is an essential step in solving equations. It involves manipulating the equation so that the variable of interest is on one side of the equation.
Let’s take the equation \(2y + 3x = 6\) and isolate \(y\). First, subtract \(3x\) from both sides to get \(2y = -3x + 6\).
Next, divide every term by 2 to solve for \(y\): \(y = -\frac{3}{2}x + 3\).
Now \(y\) is isolated, which makes it easy to identify the slope and y-intercept. Isolating variables is a key skill in algebra and is used in many different types of equations.
Knowing how to isolate a variable will help you solve and understand various math problems!