/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 41 An equation that defines \(y\) a... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 3: Problem 41

An equation that defines \(y\) as a function fof \(x\) is given. (a) Solve for \(y\) in terms of \(x,\) and \(r e-\) place \(y\) with the function notation \(f(x) .\) (b) Find \(f(3) .\) See Example 6. $$ 4 x-3 y=8 $$

Short Answer

Expert verified
The function is \(f(x) = \frac{4x - 8}{3}\). The value of \(f(3)\) is \(\frac{4}{3}\).

Step by step solution

01

Isolate the term with y

Start with the given equation: \[4x - 3y = 8\]Subtract 4x from both sides to isolate the term with y:\[-3y = 8 - 4x\]
02

Solve for y

Divide both sides of the equation by -3 to solve for y:\[y = \frac{8 - 4x}{-3}\]Simplify the expression:\[y = \frac{-8 + 4x}{3} = \frac{4x - 8}{3}\]
03

Write y as a function of x

Replace y with function notation \(f(x)\):\[f(x) = \frac{4x - 8}{3}\]
04

Substitute x = 3 into the function

To find \(f(3)\), substitute the value of x into the function:\[f(3) = \frac{4(3) - 8}{3}\]Simplify the expression:\[f(3) = \frac{12 - 8}{3} = \frac{4}{3}\]

Unlock Step-by-Step Solutions & Ace Your Exams!

  • Full Textbook Solutions

    Get detailed explanations and key concepts

  • Unlimited Al creation

    Al flashcards, explanations, exams and more...

  • Ads-free access

    To over 500 millions flashcards

  • Money-back guarantee

    We refund you if you fail your exam.

Over 30 million students worldwide already upgrade their learning with 91影视!

Key Concepts

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

Function Notation
Function notation is a way to write equations that makes it clear a variable (usually y) depends on another variable (usually x). Instead of writing an equation like y = 2x + 3, we write it as f(x) = 2x + 3. This tells us that y is a function of x.

Function notation has several advantages:
  • It is more concise and easier to read.
  • It explicitly shows the input-output relationship between variables.
  • It can handle more complex relationships and multiple variables.
For example, consider the equation y = (4x - 8)/3. Using function notation, we write it as f(x) = (4x - 8)/3, which clearly shows y is a function of x.
Solving for y
Solving for y in an equation means isolating the y variable on one side of the equation. This often involves algebraic manipulation, like addition, subtraction, multiplication, or division.

For the equation 4x - 3y = 8, we can follow these steps:
  • Subtract 4x from both sides: -3y = 8 - 4x
  • Divide everything by -3: y = (8 - 4x)/-3
  • Simplify the negative signs: y = (4x - 8)/3
Now, we've isolated y and simplified the equation. Following these steps helps express y in terms of x.
Substitution Method
The substitution method involves replacing a variable with a specific value or another expression. This technique is commonly used in algebra to find the value of one variable given another.

For example, to find f(3) in the equation f(x) = (4x - 8)/3, we substitute x with 3:
  • Replace x with 3: f(3) = (4(3) - 8)/3
  • Simplify the expression: f(3) = (12 - 8)/3
  • Finish the calculation: f(3) = 4/3
The substitution method provides a straightforward way to evaluate functions at specific points.
Simplification
Simplification involves reducing an equation or expression to its simplest form. This often makes solving equations or evaluating functions easier.

When simplifying, look for ways to combine like terms, factor common terms, or reduce fractions. For instance, with the equation (8 - 4x)/-3, we can simplify:
  • Distribute the negative sign: -8 + 4x
  • Write it as: (4x - 8)/3
Simplification helps make equations more manageable and reveals the underlying relationships between variables.

One App. One Place for Learning.

All the tools & learning materials you need for study success - in one app.

Get started for free

Most popular questions from this chapter

Find the \(x\) - and \(y\) -intercepts. Then graph each equation. $$ 4 y=3 x $$

The table represents a linear function. (a) What is \(f(2) ?\) (b) If \(f(x)=-2.5,\) what is the value of \(x ?\) (c) What is the slope of the line? (d) What is the \(y\) -intercept of the line? (e) Using your answers from parts (c) and (d), write an equation for \(f(x)\) $$ \begin{array}{c|c} x & y=f(x) \\ \hline 0 & 3.5 \\ \hline 1 & 2.3 \\ \hline 2 & 1.1 \\ \hline 3 & -0.1 \\ \hline 4 & -1.3 \\ \hline 5 & -2.5 \end{array} $$

For each situation, (a) write an equation in the form \(y=m x+b,(b)\) find and interpret the ordered pair associated with the equation for \(x=5,\) and \((c)\) answer the question. A membership in the Midwest Athletic Club costs \(\$ 99,\) plus \(\$ 41\) per month. (Source: Midwest Athletic Club.) Let \(x\) represent the number of months and \(y\) represent the cost. How much does the first year's membership cost?

An equation that defines \(y\) as a function fof \(x\) is given. (a) Solve for \(y\) in terms of \(x,\) and \(r e-\) place \(y\) with the function notation \(f(x) .\) (b) Find \(f(3) .\) See Example 6. Fill in each blank with the correct response. The equation \(2 x+y=4\) has a straight _____ as its graph. One point that lies on the graph is \((3\),____ ).If we solve the equation for \(y\) and use function notation, we obtain \(f(x)=\) _____ . For this function, \(f(3)=\) _____ ,meaning that the point ( ____,_____ ) lies on the graph of the function.

For each function, find \((a) f(2)\) and \((b) f(-1) .\) See Examples 4 and \(5 .\) $$ f=\\{(2,5),(3,9),(-1,11),(5,3)\\} $$
See all solutions

Recommended explanations on Math Textbooks

Logic and Functions

Read Explanation

Probability and Statistics

Read Explanation

Mechanics Maths

Read Explanation

Pure Maths

Read Explanation

Geometry

Read Explanation

Decision Maths

Read Explanation
View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

Study anywhere. Anytime. Across all devices.