/*! 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 3 Solve each equation. Check your ... [FREE SOLUTION] | 91影视

91影视

Log out

Learning Materials

Features

Discover

Chapter 8: Problem 3

Solve each equation. Check your solution. $$2(a-2)=3(a-5)$$

Short Answer

Expert verified
The solution is \(a = 11\).

Step by step solution

01

Expand Both Sides

First, we need to simplify both sides of the equation by applying the distributive property.For the left side: \[2(a-2) = 2a - 4\]For the right side:\[3(a-5) = 3a - 15\]So the equation becomes:\[2a - 4 = 3a - 15\]
02

Isolate Variable Terms

To isolate terms with the variable on one side, subtract \(2a\) from both sides:\[2a - 4 - 2a = 3a - 15 - 2a\]This simplifies to: \[-4 = a - 15\]
03

Solve for Variable

To solve for \(a\), add 15 to both sides of the equation:\[-4 + 15 = a - 15 + 15\]Simplify the equation:\[11 = a\]
04

Check Solution

Substitute \(a = 11\) back into the original equation to verify the solution.Original equation:\[2(a-2) = 3(a-5)\]Substituting 11:\[2(11-2) = 3(11-5)\]Calculate both sides:\[2 \times 9 = 3 \times 6\]\[18 = 18\]Since both sides of the equation are equal, \(a = 11\) is the correct solution.

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 allows us to simplify expressions. When you see an equation where a number is multiplied by a binomial, like \( 2(a-2) \), you apply this property. This means you distribute, or multiply, the number outside the parentheses by each term inside.
For example, in \( 2(a-2) \), you multiply 2 by both \( a \) and \( -2 \), resulting in \( 2a - 4 \).
Similarly, for \( 3(a-5) \), you distribute the 3 to get \( 3a - 15 \).
After you apply the distributive property, the equation is in a format that's easier to work with for solving.
Variable Isolation
Once you've simplified the equation using the distributive property, the next step is to isolate the variable. This means getting the term with the variable on one side of the equation and the constant numbers on the other.
In our example, after using the distributive property, you have \( 2a - 4 = 3a - 15 \).
  • First, move all terms with \( a \) to one side by subtracting \( 2a \) from both sides.
  • This simplifies the equation to \( -4 = a - 15 \).
  • Next, get \( a \) alone by adding 15 to both sides.
This results in \( a = 11 \).
Variable isolation is crucial because it helps find the value of unknowns in equations.
Checking Solutions
After finding a potential solution, it鈥檚 essential to check if it truly satisfies the original equation. This ensures that no errors were made in the calculations.
  • Return to the original problem: \( 2(a-2) = 3(a-5) \).
  • Substitute the found value, \( a = 11 \), back into the equation.
  • The left side becomes \( 2(11-2) = 2 \times 9 \), which equals 18.
  • The right side becomes \( 3(11-5) = 3 \times 6 \), which also equals 18.
Since both sides match, we confirm our solution is correct.
Checking your work by substituting the solution back into the equation is always a good practice, helping to catch any mistakes made during solving.

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

Solve each inequality and check your solution. Then graph the solution on a number line. $$-2(k+1) \geq 16$$

Express each number in scientific notation. $$-0.00203$$

Solve each inequality and check your solution. Then graph the solution on a number line. $$c-1<3 c+5$$

Express each ratio as a unit rate. \(\$ 5\) for 2 loaves of bread

Write \(\frac{1}{200}\) as a percent.
See all solutions

Recommended explanations on Math Textbooks

Mechanics Maths

Read Explanation

Theoretical and Mathematical Physics

Read Explanation

Calculus

Read Explanation

Geometry

Read Explanation

Pure Maths

Read Explanation

Logic and Functions

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.