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Chapter 2: Problem 39

Determine the slope and the \(y\) -intercept. $$ 4 x+5 y=8 $$

Short Answer

Expert verified
Slope: \(-\frac{4}{5}\), \(y\)-intercept: \(\frac{8}{5}\)

Step by step solution

01

Rewrite the equation

Rewrite the equation in slope-intercept form, which is given by \(y = mx + b\), where \(m\) is the slope and \(b\) is the \(y\)-intercept.
02

Isolate \(y\)

Start with the equation: \(4x + 5y = 8\). Isolate \(y\) by subtracting \(4x\) from both sides: \(5y = -4x + 8\).
03

Solve for \(y\)

Divide every term by 5 to solve for \(y\): \(y = -\frac{4}{5}x + \frac{8}{5}\). Now the equation is in the form \(y = mx + b\).
04

Identify the slope

In the equation \(y = -\frac{4}{5}x + \frac{8}{5}\), the coefficient of \(x\) is the slope \(m\). Thus, \(m = -\frac{4}{5}\).
05

Identify the \(y\)-intercept

In the same equation, the constant term is the \(y\)-intercept \(b\). Thus, \(b = \frac{8}{5}\).

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Key Concepts

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

linear equations
Linear equations are mathematical statements that establish a relationship between two variables, usually x and y, in the form of a straight line when graphed.
They can be represented in various forms, but one of the most recognizable is the slope-intercept form.
A linear equation might look something like this: 4x + 5y = 8.

Linear equations are called 'linear' because their graph is always a straight line. The 'linear' aspect references the degree of the equation, which is 1.
Each term either has a constant or a variable raised to the first power.
  • Example: y = 2x + 3
  • Linear: Neither variable is squared or higher.
Linear equations are fundamental in algebra and are used to model many real-world situations.
slope-intercept form
The slope-intercept form of a linear equation is a very convenient way to express a line.
It鈥檚 written as: \( y = mx + b \) where:
  • \( m \) is the slope.
  • \( b \) is the y-intercept.
To convert a linear equation to this form, you need to isolate y on one side of the equation.
Let's take the example from the exercise:
Start with: 4x + 5y = 8

Step-by-Step:
  • Subtract 4x from both sides: 5y = -4x + 8
  • Divide every term by 5: y = - \(\frac{4}{5}\) x + \(\frac{8}{5}\)
Now, the equation is in slope-intercept form: y = - \( \frac{4}{5} \) x + \( \frac{8}{5} \)
You can clearly see the slope (m) and y-intercept (b).
solving for y
Solving for y means isolating y on one side of the equation.
This results in the slope-intercept form, making it easy to understand the line's behavior.
Here鈥檚 the step-by-step process using the exercise example:

Starting Equation: 4x + 5y = 8

Step 1: Subtract the x-term from both sides to start isolating y.
4x + 5y - 4x = 8 - 4x
5y = -4x + 8

Step 2: Divide every term by the coefficient of y (which is 5).
y = - \( \frac{4}{5} \) x + \( \frac{8}{5} \)
Now y is isolated and the equation is in slope-intercept form.

To sum up:
  • Subtract the x-term from both sides to move it.
  • Divide by the coefficient of y to finish isolating y.
This method allows you to clearly see the slope and the y-intercept.

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