Problem 120
Question
Exercises \(120-122\) will help you prepare for the material covered in the first section of the next chapter. In each exercise, determine whether the given number is a solution of the equation. $$-\frac{1}{2}=x-\frac{2}{3} ; \frac{1}{6}$$
Step-by-Step Solution
Verified Answer
\(\frac{1}{6}\) is a solution of the equation \(-\frac{1}{2}=x-\frac{2}{3}\).
1Step 1: Identify the Equation and Potential Solution
The equation provided is \(-\frac{1}{2}=x-\frac{2}{3}\) and the number to verify as a solution is \(\frac{1}{6}\).
2Step 2: Substitute the Potential Solution into Equation
To verify if \(\frac{1}{6}\) is indeed a solution, it is substituted in place of \(x\) in the equation. This gives \(-\frac{1}{2}=\frac{1}{6}-\frac{2}{3}\).
3Step 3: Simplify the Equation
Next, simplify the right side of the equation. This will result in \(-\frac{1}{2}=-\frac{1}{2}\).
4Step 4: Verify the Equality
Since both sides of the equation are equal after substituting the given value, it verifies that \(\frac{1}{6}\) is indeed a solution to the equation.
Key Concepts
Algebraic SolutionsEquation VerificationSubstitution Method
Algebraic Solutions
When working through algebra problems, finding algebraic solutions is often the primary goal. An algebraic solution refers to a value, which when substituted into an equation, satisfies the equation, making it a true statement. For example, if we have the equation \( -\frac{1}{2} = x - \frac{2}{3} \), we are looking for a number that, when placed in place of \( x \), will ensure that both sides of the equation are equal.
Determining the correct value involves a combination of operations such as addition, subtraction, multiplication, and division. In the case of our exercise, the solution is determined step by step—beginning with identifying the equation, substituting the potential solution, and then simplifying. When \( \frac{1}{6} \) is substituted for \( x \), we find that the equation holds true, thus \( \frac{1}{6} \) is indeed the algebraic solution.
Determining the correct value involves a combination of operations such as addition, subtraction, multiplication, and division. In the case of our exercise, the solution is determined step by step—beginning with identifying the equation, substituting the potential solution, and then simplifying. When \( \frac{1}{6} \) is substituted for \( x \), we find that the equation holds true, thus \( \frac{1}{6} \) is indeed the algebraic solution.
Equation Verification
The process of equation verification is critical to confirm whether a value is a solution to an equation. After obtaining a potential solution, there's a need for proof that this value is correct. That's where verification comes into play.
To verify, you replace the variable in the equation with the potential solution and simplify the expression. If the left and right sides of the equation remain equal after this substitution, then the potential solution is a verified solution of the equation. In our textbook example, after substituting \( \frac{1}{6} \) for \( x \) and performing the necessary arithmetic, it led to an identity \( -\frac{1}{2} = -\frac{1}{2} \), thus verifying that \( \frac{1}{6} \) is indeed a solution.
To verify, you replace the variable in the equation with the potential solution and simplify the expression. If the left and right sides of the equation remain equal after this substitution, then the potential solution is a verified solution of the equation. In our textbook example, after substituting \( \frac{1}{6} \) for \( x \) and performing the necessary arithmetic, it led to an identity \( -\frac{1}{2} = -\frac{1}{2} \), thus verifying that \( \frac{1}{6} \) is indeed a solution.
Tips for Successful Verification
- Check your arithmetic carefully to avoid errors.
- Remember to simplify both sides of the equation completely before making a conclusion.
- Ensure to maintain the balance of the equation by performing the same operation on both sides when needed.
Substitution Method
The substitution method is a powerful tool in algebra, especially when verifying solutions to equations or solving systems of equations. This method involves replacing variables with their potential solutions to test for correctness.
In the context of our example, we use the substitution method by taking the value given \( \frac{1}{6} \) and substituting it directly into the original equation in place of \( x \). The main steps are substituting, simplifying, and then comparing both sides of the equation.
In the context of our example, we use the substitution method by taking the value given \( \frac{1}{6} \) and substituting it directly into the original equation in place of \( x \). The main steps are substituting, simplifying, and then comparing both sides of the equation.
Advantages of Substitution
- It's straightforward and systematic, reducing the chances of error.
- This method can be used for both linear and non-linear equations.
- Substitution is very applicable for equations where the value of one variable is already known or can be easily isolated.
Other exercises in this chapter
Problem 119
Perform the indicated operations. Begin by performing operations in parentheses. $$\left(\frac{1}{2}-\frac{1}{3}\right) \div \frac{5}{8}$$
View solution Problem 120
Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement. If \(a\) and \(b\) are n
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Use a calculator to find a decimal approximation for each irrational number, correct to three decimal places. Between which two integers should you graph each o
View solution Problem 120
Perform the indicated operations. Begin by performing operations in parentheses. $$\left(\frac{1}{2}+\frac{1}{4}\right) \div\left(\frac{1}{2}+\frac{1}{3}\right)
View solution