/*! 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 9 In each set of three equations, ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 4: Problem 9

In each set of three equations, two equations are equivalent. Find them and explain how you know they are equivalent. a. i. \(y=14-2(x-5)\) b. i. \(y=-13+4(x+2)\) (a) ii. \(y=30-2(x+3)\) ii. \(y=10+3(x-5)\) iii. \(y=-12+2(x-5)\) iii. \(y=-25+4(x+5)\) c. i. \(y=5+5(x-8)\) d. i. \(y=-16+6(x+5)\) ii. \(y=9+5(x+8)\) ii. \(y=8+6(x-5)\) iii. \(y=94+5(x-9)\) iii. \(y=44+6(x-5)\)

Short Answer

Expert verified
Sets a, c, and d have equivalent pairs: a. i. == a. ii.; c. ii. == c. iii.; d. i. == d. iii.

Step by step solution

01

Expand Equation a. i.

Start with the equation \( y = 14 - 2(x-5) \). Distribute the \(-2\) across the expression \((x-5)\). \\[ y = 14 - 2x + 10 \] \Combine like terms to get: \\[ y = 24 - 2x \]
02

Expand Equation a. ii.

Now take the equation \( y = 30 - 2(x+3) \) and distribute the \(-2\) across \((x+3)\). \\[ y = 30 - 2x - 6 \] \Combine like terms: \\[ y = 24 - 2x \]
03

Compare Equations a. i. and a. ii.

The expanded forms of both equations a. i. and a. ii. are \( y = 24 - 2x \), indicating they are equivalent.
04

Expand Equation b. i.

Expand \( y = -13 + 4(x+2) \): \\[ y = -13 + 4x + 8 \] \Combine like terms: \\[ y = 4x - 5 \]
05

Expand Equation b. ii.

Expand \( y = 10 + 3(x-5) \): \\[ y = 10 + 3x - 15 \] \Combine like terms: \\[ y = 3x - 5 \]
06

Compare Equations b. i. and b. ii.

The expanded forms \( y = 4x - 5 \) (b. i.) and \( y = 3x - 5 \) (b. ii.) are not equivalent. Calculate the other equation for comparison.
07

Expand Equation b. iii.

Expand \( y = -12 + 2(x-5) \): \\[ y = -12 + 2x - 10 \] \Combine like terms: \\[ y = 2x - 22 \]
08

Verify Equation Pair for Set b

None of the given equations b. i., b. ii., and b. iii. are equivalent as identified by the distinct results of expansions.
09

Expand Equation c. i.

Expand \( y = 5 + 5(x-8) \): \\[ y = 5 + 5x - 40 \] \Combine like terms: \\[ y = 5x - 35 \]
10

Expand Equation c. ii.

Expand \( y = 9 + 5(x+8) \): \\[ y = 9 + 5x + 40 \] \Combine like terms: \\[ y = 5x + 49 \]
11

Compare Equations c. i. and c. ii.

Equations c. i. and c. ii. are not equivalent. Check the third equation.
12

Expand Equation c. iii.

Expand \( y = 94 + 5(x-9) \): \\[ y = 94 + 5x - 45 \] \Combine like terms: \\[ y = 5x + 49 \] \It matches the expanded form of c. ii.
13

Verify Equation Pair for Set c

The equations c. ii. \( y = 5x + 49 \) and c. iii. \( y = 5x + 49 \) are equivalent.
14

Expand Equation d. i.

Expand \( y = -16 + 6(x+5) \): \\[ y = -16 + 6x + 30 \] \Combine like terms: \\[ y = 6x + 14 \]
15

Expand Equation d. ii.

Expand \( y = 8 + 6(x-5) \): \\[ y = 8 + 6x - 30 \] \Combine like terms: \\[ y = 6x - 22 \]
16

Compare Equations d. i. and d. ii.

Equations d. i. and d. ii. are not equivalent. Check the third equation.
17

Expand Equation d. iii.

Expand \( y = 44 + 6(x-5) \): \\[ y = 44 + 6x - 30 \] \Combine like terms: \\[ y = 6x + 14 \] \It matches the expanded form of d. i.
18

Verify Equation Pair for Set d

The equations d. i. \( y = 6x + 14 \) and d. iii. \( y = 6x + 14 \) are equivalent.

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.

Distributive Property
The distributive property is a fundamental concept in algebra that simplifies expressions and equations. We often encounter it when we need to multiply a single term by more than one term inside parentheses. This property can be expressed in the general form as: \( a(b + c) = ab + ac \).
In simpler terms, you take each term inside the parenthesis and multiply it by the term outside the parenthesis.
  • For example, in the equation \( y = 14 - 2(x-5) \), we use the distributive property by multiplying \(-2\) with both \(x\) and \(-5\).
  • This leads to \( y = 14 - 2x + 10 \).
By applying the distributive property correctly, we make it easier to solve or compare equations. This is particularly useful when working with linear equations, as we'll see in the following sections.
Combining Like Terms
Once you've used the distributive property, you'll often end up with an expression that has similar terms. Combining like terms makes equations simpler and easier to solve. Like terms are terms that have the same variables raised to the same power.
  • In the example \( y = 14 - 2x + 10 \), the terms \( 14 \) and \( 10 \) are like terms because they are constants.
  • These combine to give us \( y = 24 - 2x \).
Aligning the coefficients or constants together helps reduce complexity when solving linear equations. It's similar to cleaning up an expression, eliminating unnecessary components and paving the way for easier solutions.
Linear Equations
Linear equations represent straight lines when graphed on a coordinate plane. They are typically written in the form \( y = mx + b \), where \( m \) is the slope and \( b \) is the y-intercept.
These equations are incredibly common across algebra, as they can describe many real-world relationships.
  • For instance, the equation \( y = 24 - 2x \) shows a linear relationship with a slope of \(-2\) and a y-intercept of \(24\).
  • Being linear means they have constant rates of change, making them simpler to analyze compared to non-linear equations.
Understanding linear equations is crucial because they serve as the foundational language of algebra, connecting various topics seamlessly.
Step-by-Step Solutions
Breaking down complex problems into clear, simple steps is essential for learning math, especially algebra. A step-by-step approach allows you to understand each part of the solution process fully.
Start with identifying the core operations required and use relevant algebraic rules like the distributive property.
  • Begin with expanding the equation using the distributive property.
  • Next, combine like terms to simplify the equation further.
  • Compare the resulting form with other equations to check for equivalence.
By going through problems methodically, you can easily resolve misunderstandings and build a strong foundation in handling more complex math problems in the future. Practicing these steps regularly enhances problem-solving skills, making math more accessible and less daunting.

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

Bryan has bought a box of biscuits for his dog, Anchor. Anchor always gets three biscuits a day. At the start of the 10th day after opening the box, Bryan counts 106 biscuits left. Let \(x\) represent the number of days after opening the box, and let \(y\) represent the number of biscuits left. a. In a graph of this situation, what is the slope? (h) b. Write a point-slope equation that models the situation. c. When will the box be empty? d. What is the real-world meaning of the \(y\)-intercept?

Write each equation in the form requested. Check your answers by graphing on your calculator. a. Write \(y=13.6(x-1902)+158.2\) in intercept form. b. Write \(y=-5.2 x+15\) in point-slope form using \(x=10\) as the first coordinate of the point.

Mini-Investigation Consider the equation \(y=10+5 x\) in intercept form. a. Factor the right side of the equation. b. Use the commutative property of addition to swap the terms inside the parentheses. c. Your result should look similar to the point-slope form of the equation. What's missing? What is the value of this missing piece? (a) d. What point could you use to write the point-slope equation in \(8 \mathrm{c}\) ? What is special about this point?

The equation of a line in point-slope form is \(y=6-3(x-6)\). a. Name the point on this line that was used to write the equation. b. Name the point on this line with an \(x\)-coordinate of 5 . c. Using the point you named in \(1 b\), write another equation of the line in point-slope form. d. Write the equation of the line in intercept form. e. Find the coordinates of the \(x\)-intercept.

At 2:00 P.M, elevator A passes the second floor of the Empire State Building going up. The table shows the floors and the times in seconds after 2:00. \begin{tabular}{|l|c|c|c|c|c|c|c|} \hline Floor \(x\) & 2 & 4 & 6 & 8 & 10 & 12 & 14 \\ \hline Time after 2:00 (s) \(y\) & 0 & \(1.3\) & \(2.5\) & \(3.8\) & 5 & \(6.3\) & \(7.5\) \\\ \hline \end{tabular} a. What is the line of fit based on Q-points for the data? b. Give a real-world meaning for the slope. (a) c. About what time will this elevator pass the 60th floor if it makes no stops? (a) d. Where will this elevator be at \(2: 00: 45\) if it makes no stops?
See all solutions

Recommended explanations on Math Textbooks

Theoretical and Mathematical Physics

Read Explanation

Logic and Functions

Read Explanation

Discrete Mathematics

Read Explanation

Statistics

Read Explanation

Pure Maths

Read Explanation

Mechanics 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.