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Chapter 1: Problem 55

Find the slope and the \(y\) -intercept of the line with the given equation. $$3 x+2 y=10$$

### Short Answer

Expert verified

The slope is \(-\frac{3}{2}\), and the \(y \, -intercept\) is \5\.

## Step by step solution

01

## - Write the equation in slope-intercept form

The slope-intercept form of a line is given by \(y = mx + c\), where \m\ is the slope and \c\ is the \(y \, -intercept\). Start by isolating \(y\) in the given equation \(3x + 2y = 10\).

02

## - Subtract 3x from both sides

To isolate \(y\), subtract \(3x\) from both sides: \(2y = -3x + 10\)

03

## - Divide by 2

Next, divide every term by \(2\) to solve for \(y\): \[y = -\frac{3}{2}x + 5\]

04

## - Identify the slope and \(y\)-intercept

In the equation \[y = -\frac{3}{2}x + 5\], the slope \(m\) is \(-\frac{3}{2}\) and the \(y \, -intercept\) \(c\) is \(5\).

## Key Concepts

These are the key concepts you need to understand to accurately answer the question.

###### linear equations

Linear equations are equations involving two variables, often written in the form of \(ax + by = c\). These equations create a straight line when plotted on a graph. For instance, the equation \(3x + 2y = 10\) is a linear equation. The main goal when working with linear equations is to explore the relationship between the variables \(x\) and \(y\). In math, it's common to rearrange these equations to reveal more information about the line's properties such as its slope and \(y\)-intercept.

###### slope-intercept form

The slope-intercept form is a specific arrangement of a linear equation. You'll see it written as \(y = mx + c\). This form is quite handy because it immediately reveals two important characteristics about the line: the slope (\(m\)) and the \(y\)-intercept (\(c\)).

- The slope, denoted by \(m\), tells us the steepness and direction of the line. If \(m\) is positive, the line rises; if \(m\) is negative, the line falls.
- The \(y\)-intercept, denoted by \(c\), is the point where the line crosses the \(y\)-axis. It's what \(y\) equals when \(x\) is \(0\).

To convert any linear equation into this form, you'll need to isolate \(y\), a process we call 'isolation of variables'.

###### isolation of variables

Isolating variables means rearranging an equation to solve for one variable in terms of others. This is key in simplifying linear equations and finding specific solutions. Let's revisit our example, \(3x + 2y = 10\):

- First, we subtract \(3x\) from both sides to get: \(2y = -3x + 10\).
- Next, we divide each term by \(2\) to isolate \(y\): \(y = -\frac{3}{2}x + 5\).

Now, the equation is in slope-intercept form. This shows the slope (\(m\)) is \(-\frac{3}{2}\) and the \(y\)-intercept (\(c\)) is \(5\). Understanding how to isolate variables will help you solve many different types of equations and understand the relationships between variables more clearly.

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