Is R Equals 26 A Solution To The Equation 2=r/13?

In the realm of mathematics, solving equations is a fundamental skill. It involves finding the value(s) of the unknown variable(s) that make the equation true. In this article, we will delve into the process of determining whether a given value is a solution to a specific equation. We will focus on the equation $2 = \frac{r}{13}$ and investigate whether $r = 26$ is indeed a solution. This exploration will not only solidify our understanding of equation-solving but also highlight the importance of verification in mathematical problem-solving.

Understanding Equations and Solutions

Before we dive into the specifics, let's establish a clear understanding of what equations and solutions are. An equation is a mathematical statement that asserts the equality of two expressions. These expressions can involve numbers, variables, and mathematical operations. The goal of solving an equation is to find the value(s) of the variable(s) that make the equation true. These values are called solutions or roots of the equation.

A solution to an equation is a value that, when substituted for the variable, results in a true statement. In other words, the left-hand side (LHS) of the equation must equal the right-hand side (RHS) after the substitution. To determine if a given value is a solution, we substitute the value into the equation and check if the equality holds.

For instance, consider the simple equation $x + 3 = 5$. If we substitute $x = 2$ into the equation, we get $2 + 3 = 5$, which is a true statement. Therefore, $x = 2$ is a solution to the equation. On the other hand, if we substitute $x = 1$, we get $1 + 3 = 5$, which is false. Thus, $x = 1$ is not a solution.

The Equation 2 = r/13

Now, let's turn our attention to the equation at hand: $2 = \frac{r}{13}$. This equation involves the variable $r$, and our task is to determine whether $r = 26$ is a solution. To do this, we will substitute $r = 26$ into the equation and see if it satisfies the equality.

Substituting $r = 26$ into the equation, we get:

$2 = \frac{26}{13}$

The next step is to simplify the right-hand side of the equation. The fraction $\frac{26}{13}$ represents the division of 26 by 13. Performing this division, we find that:

$\frac{26}{13} = 2$

Now, we can rewrite the equation with the simplified right-hand side:

$2 = 2$

This statement is clearly true. The left-hand side (2) is equal to the right-hand side (2). Therefore, we can conclude that $r = 26$ is indeed a solution to the equation $2 = \frac{r}{13}$.

Solving the Equation Algebraically

While we have verified that $r = 26$ is a solution, let's also solve the equation algebraically to further solidify our understanding. Solving an equation algebraically involves isolating the variable on one side of the equation using mathematical operations.

Our equation is $2 = \frac{r}{13}$. To isolate $r$, we need to undo the division by 13. The inverse operation of division is multiplication. Therefore, we will multiply both sides of the equation by 13.

Multiplying both sides by 13, we get:

$2 \cdot 13 = \frac{r}{13} \cdot 13$

On the left-hand side, we have $2 \cdot 13$, which equals 26. On the right-hand side, the multiplication by 13 cancels out the division by 13, leaving us with just $r$:

$26 = r$

This result confirms that $r = 26$ is the solution to the equation. We have arrived at the same conclusion through both substitution and algebraic manipulation.

Importance of Verification

The process of verifying solutions is crucial in mathematics. It helps us ensure that our answers are correct and that we haven't made any errors in our calculations. Simply solving an equation doesn't guarantee that the answer is correct; we must always check our solution by substituting it back into the original equation.

Verification is especially important when dealing with more complex equations, where errors are more likely to occur. It also helps us catch extraneous solutions, which are values that satisfy a transformed equation but not the original equation. Extraneous solutions can arise when we perform operations that are not reversible, such as squaring both sides of an equation.

In the case of our equation, $2 = \frac{r}{13}$, verification was straightforward. However, in more complex scenarios, the verification process may involve more steps and careful attention to detail. By making verification a habit, we can increase our confidence in our solutions and improve our problem-solving skills.

Conclusion

In this article, we investigated whether $r = 26$ is a solution to the equation $2 = \frac{r}{13}$. Through both substitution and algebraic manipulation, we confirmed that $r = 26$ is indeed a solution. We substituted $r = 26$ into the equation and found that it resulted in a true statement, $2 = 2$. We also solved the equation algebraically by multiplying both sides by 13, which yielded the solution $r = 26$.

Furthermore, we emphasized the importance of verification in mathematical problem-solving. Verification helps us ensure the accuracy of our solutions and catch any potential errors. By substituting our solution back into the original equation, we can confirm whether it satisfies the equality.

This exploration has reinforced our understanding of equations, solutions, and the crucial role of verification in mathematics. By mastering these concepts, we can confidently tackle a wide range of mathematical problems and build a strong foundation for further learning.

To further illustrate the process of verifying solutions, let's consider a few more examples.

Example 1

Equation: $3x - 5 = 10$

Proposed solution: $x = 5$

To verify if $x = 5$ is a solution, we substitute it into the equation:

$3(5) - 5 = 10$

Simplifying the left-hand side, we get:

$15 - 5 = 10$

$10 = 10$

Since the left-hand side equals the right-hand side, $x = 5$ is a solution to the equation.

Example 2

Equation: $x^2 - 4 = 0$

Proposed solution: $x = -2$

Substituting $x = -2$ into the equation:

$(-2)^2 - 4 = 0$

Simplifying the left-hand side:

$4 - 4 = 0$

$0 = 0$

Thus, $x = -2$ is a solution to the equation.

Example 3

Equation: $\frac{1}{x} + 2 = 5$

Proposed solution: $x = \frac{1}{3}$

Substituting $x = \frac{1}{3}$ into the equation:

$\frac{1}{\frac{1}{3}} + 2 = 5$

Simplifying the left-hand side:

$3 + 2 = 5$

$5 = 5$

Therefore, $x = \frac{1}{3}$ is a solution to the equation.

These examples demonstrate the consistent process of substituting the proposed solution into the equation and verifying if it results in a true statement. This practice is fundamental to ensuring the accuracy of our mathematical solutions.

When solving and verifying solutions to equations, there are several common mistakes that students often make. Being aware of these pitfalls can help you avoid errors and improve your problem-solving accuracy.

1. Incorrect Substitution

One of the most common mistakes is substituting the value incorrectly into the equation. This can involve substituting the value for the wrong variable, miscopying the value, or making errors in the substitution process itself. To avoid this, double-check your substitution and ensure that you are replacing the correct variable with the given value.

2. Arithmetic Errors

Arithmetic errors, such as mistakes in addition, subtraction, multiplication, or division, can lead to incorrect results. These errors can occur both when simplifying the equation and when verifying the solution. To minimize arithmetic errors, work carefully, show your steps, and use a calculator if necessary.

3. Incorrect Order of Operations

The order of operations (PEMDAS/BODMAS) is crucial in mathematics. Failing to follow the correct order of operations can lead to incorrect simplification and verification. Remember to perform operations in the following order: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).

4. Forgetting to Distribute

When an equation involves parentheses, it's essential to distribute any factors outside the parentheses to all terms inside. Forgetting to distribute can lead to an incorrect equation and, consequently, an incorrect solution. For example, in the equation $2(x + 3) = 10$, you must distribute the 2 to both $x$ and 3.

5. Not Simplifying Completely

Before verifying a solution, it's important to simplify both sides of the equation as much as possible. Failure to simplify completely can make it difficult to see if the equation is true after substitution. For instance, if you have an equation with fractions, simplify the fractions before substituting the value.

6. Ignoring Extraneous Solutions

As mentioned earlier, extraneous solutions can arise when performing non-reversible operations. It's crucial to check all solutions in the original equation to identify and discard any extraneous solutions. This is especially important when dealing with equations involving radicals or rational expressions.

7. Assuming the Answer is Correct

Perhaps the biggest mistake is assuming that the answer is correct without verifying it. Verification is the only way to be sure that your solution is accurate. Always take the time to substitute your solution back into the original equation and check if it satisfies the equality.

By being mindful of these common mistakes and taking steps to avoid them, you can significantly improve your accuracy in solving and verifying equations.

In conclusion, determining whether a given value is a solution to an equation is a fundamental skill in mathematics. It involves substituting the value into the equation and checking if the equality holds true. We have demonstrated this process with the equation $2 = \frac{r}{13}$ and confirmed that $r = 26$ is indeed a solution. Furthermore, we have emphasized the importance of verification in mathematical problem-solving and discussed common mistakes to avoid.

By mastering the concepts and techniques presented in this article, you can enhance your ability to solve equations accurately and confidently. Remember to always verify your solutions and be mindful of potential errors. With practice and diligence, you can excel in mathematical problem-solving and build a strong foundation for further learning.