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Chapter 1: Problem 58

Decide whether the given number is a solution of the equation. \(6 x+2(x+3)=14 ; 2\)

Short Answer

Expert verified
No, 2 is not a solution.

Step by step solution

01

Substitute the number into the equation

Replace the variable x in the equation with the given number 2. The equation is now: 6(2) + 2(2 + 3) = 14.
02

Simplify the inside of the parentheses

First, simplify the expression inside the parentheses: 2 + 3 = 5 So, the equation becomes: 6(2) + 2(5) = 14.
03

Multiply to simplify further

Carry out the multiplication: 6(2) = 12 2(5) = 10 Substitute these values back into the equation: 12 + 10 = 14.
04

Add the results

Add the simplified values: 12 + 10 = 22 Compare the result with the right-hand side of the original equation: 22 鈮� 14 Therefore, the equation 6x + 2(x + 3) = 14 is not satisfied by x = 2.

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Key Concepts

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

Substitution in equations
Substitution is a powerful technique in algebra. It's often one of the first steps in solving equations. You substitute a specific number for a variable. In our exercise, we replace the variable \( x \) with 2. This helps us verify if the number satisfies the equation. By substituting \( x = 2 \), our initial equation \[ 6x + 2(x + 3) = 14 \] changes. It becomes: \[ 6(2) + 2(2 + 3) = 14 \].
This lets us work with concrete numbers instead of variables. Remember, always replace the variable consistently throughout the equation.
Simplifying algebraic expressions
Simplifying expressions is a crucial part of algebra. It makes equations easier to understand and solve. After substitution, we simplify inside the parentheses first. In our example, we simplify \( 2 + 3 \). This equals 5. So, our equation becomes: \[ 6(2) + 2(5) = 14 \].
Then, perform any other simple arithmetic inside the expression. This reduced form helps in further calculations. Always simplify step by step, efficiently manage complexity.
Verifying solutions to equations
Verifying a solution means checking if a substituted value satisfies the equation. After substituting and simplifying, our example stands at: \[ 6(2) + 2(5) = 14 \].
We carry out the multiplication next. Namely, \( 6(2) = 12 \) and \( 2(5) = 10 \). Substituting back, we get: \[ 12 + 10 = 22 \].
We then compare this result (22) with the right-hand side of our original equation (14). Since \[ 22 ot= 14 \], the number 2 does not satisfy the equation. This shows how verifying can confirm whether a solution is correct or not.
Basic arithmetic operations in algebra
Basic arithmetic lays the foundation of algebraic operations. In the context of our example, we use addition, multiplication, and comparison.
First, arithmetic inside parentheses: \( 2 + 3 = 5 \).
Next, multiplication: \[ 6(2) = 12 \] and \[ 2(5) = 10 \]. Lastly, addition: \[ 12 + 10 = 22 \].
These operations simplify and transform expressions, making complex problems more approachable. Always ensure to carry these operations methodically to avoid errors.

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