Problem 72 Solve and check each equation. $... [FREE SOLUTION]

Chapter 3: Problem 72

Solve and check each equation. $$20=5 r-3+2(9-3 r)$$

Short Answer

Expert verified

The solution to the equation $20=5r-3+2(9-3r)$ is $r = -5$. This was found by first distributing the constant 2, then combining like terms, isolating the variable, and finally checking the solution by plugging it back into the original equation.

Step by step solution

Distribute the constant

Apply the distributive property to the equation by multiplying `2` with the terms inside the parentheses: $ 2(9 - 3r) = 2 \cdot 9 - 2 \cdot 3r$. So, the equation becomes: \[20 = 5r - 3 + 18 - 6r.\]

Combine like terms

Combine like terms by adding `5r` and `-6r`, and by adding `-3` and `18`. The equation then becomes: \[20 = -r + 15.\]

Isolate the variable

To isolate `r`, subtract `15` from both sides of the equation: \[20 - 15 = -r.\] Thus, we get the equation \[5 = -r.\]

Solve for r

To solve for `r`, we multiply both sides by `-1`, so we get \[r = -5.\]

Check the solution

To check the solution, substitute `-5` into the original equation and determine if the left side equals the right side. \[20 = 5(-5) - 3 + 2(9 - 3(-5)).\] Simplify both sides of the equation: \[20 = -25 - 3 + 18 + 30.\] This simplifies to \[20 = 20.\] Since the left side equals the right side, the solution is correct, and the equation holds true for `r = -5`.

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.

Understanding the Distributive Property

The distributive property is a fundamental concept in algebra that helps to simplify equations, especially when it involves terms within parentheses. In this context, we are dealing with the expression $2(9 - 3r)$. Here鈥檚 how we apply the distributive property:

Multiply the number outside the parentheses (2) by each term inside the parentheses (9 and $-3r$).
This means we perform: $2 \cdot 9$ and $2 \cdot -3r$.

This yields the expression $18 - 6r$.
By applying the distributive property, we transform the equation into an easier form to manage: $20 = 5r - 3 + 18 - 6r$.
Mastering the distributive property allows you to handle equations with ease, making complex expressions more straightforward to solve.

Combining Like Terms

Combining like terms is an essential step when simplifying algebraic equations. Like terms are terms that contain the same variable raised to the same power. In our case, we have $5r - 6r$ and $-3 + 18$.

For the variable terms $5r - 6r$, subtract the coefficients to find: $-r$.
For the constant terms $-3 + 18$, simply add them up to get: 15.

This combination simplifies the equation to $20 = -r + 15$.
Combining like terms reduces clutter, helping to reveal a clear pathway to solving the equation. It鈥檚 crucial in equation simplification and ensures all equivalents are neatly organized.

Variable Isolation in Algebra

Isolating the variable is a critical skill for solving equations in algebra. When we aim to isolate $r$, our goal is to have $r$ alone on one side of the equation. Here鈥檚 how it鈥檚 done:

Start with: $20 = -r + 15$.
Subtract 15 from both sides to move the constant term: $20 - 15 = -r$.
This results in: $5 = -r$.

Getting $-r$ means we have $r$ almost isolated. We multiply both sides by $-1$ to change $-r$ to $r$.
The manipulation leads us to the solution: $r = -5$.
Understanding variable isolation is vital as it鈥檚 a common final step in solving algebraic equations.

Importance of Checking Solutions

Once we find a solution, checking it is just as important to confirm its validity. Substituting back into the original equation provides reassurance. Here鈥檚 the process:

Plug $r = -5$ back into the original equation $20 = 5r - 3 + 2(9 - 3r)$.
It becomes: $20 = 5(-5) - 3 + 2(9 - 3(-5))$.
Simplify both sides: $20 = -25 - 3 + 18 + 30$.
After simplification: $20 = 20$.

Since both sides of the equation are equal, $r = -5$ is verified as the correct solution.
Checking solutions is a vital verification step ensuring accuracy in mathematical problem-solving.

One App. One Place for Learning.

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

Get started for free

Recommended explanations on Math Textbooks

View all explanations

What do you think about this solution?

We value your feedback to improve our textbook solutions.

91影视

Solve and check each equation. $$20=5 r-3+2(9-3 r)$$

Short Answer

Step by step solution

Distribute the constant

Combine like terms

Isolate the variable

Solve for r

Check the solution

Key Concepts

Understanding the Distributive Property

Combining Like Terms

Variable Isolation in Algebra

Importance of Checking Solutions

One App. One Place for Learning.

Most popular questions from this chapter

Recommended explanations on Math Textbooks

Pure Maths

Calculus

Applied Mathematics

Logic and Functions

Discrete Mathematics

Mechanics Maths

Study anywhere. Anytime. Across all devices.