Solving (2/3)y + 15 = 9 A Step-by-Step Guide With Justifications

In the realm of mathematics, solving equations is a fundamental skill. It allows us to unravel the mysteries behind mathematical expressions and find the values that satisfy them. In this comprehensive guide, we will embark on a journey to dissect and solve the equation $\frac{2}{3} y+15=9$, providing a step-by-step explanation of each operation and its justification. Understanding the properties of equality is essential for solving algebraic equations. The properties of equality allow us to manipulate equations while maintaining their balance and finding the correct solution. This guide provides a detailed walkthrough for solving the equation, emphasizing the importance of each step and the mathematical principles behind it.

To isolate the term containing the variable y, we need to eliminate the constant term on the left side of the equation. The subtraction property of equality states that subtracting the same value from both sides of an equation maintains the equality. This property is a cornerstone of algebraic manipulation, allowing us to simplify equations without altering their fundamental balance. By subtracting 15 from both sides, we effectively neutralize the +15 on the left, paving the way to isolate the term with y. This step is crucial because it begins the process of isolating the variable, which is the primary goal in solving any equation. Imagine an equation as a balanced scale; whatever you do to one side, you must do to the other to maintain equilibrium. Subtracting 15 from both sides keeps the equation balanced and moves us closer to the solution.

$\frac{2}{3} y+15=9$

Subtract 15 from both sides:

$\frac{2}{3} y+15 - 15 = 9 - 15$

Simplify:

$\frac{2}{3} y = -6$

This step is justified by the subtraction property of equality, which allows us to subtract the same value from both sides of an equation without changing its solution. Subtracting 15 from both sides isolates the term with the variable y.

Now that we have $\frac{2}{3}y = -6$, our goal is to isolate y completely. To do this, we need to eliminate the coefficient $\frac{2}{3}$ that is multiplying y. The multiplication property of equality comes into play here. This property states that multiplying both sides of an equation by the same non-zero value maintains the equality. This is another fundamental principle in algebra, allowing us to scale both sides of the equation proportionally without disrupting its balance. To eliminate the fraction, we multiply both sides by its reciprocal, which is $\frac{3}{2}$. The reciprocal of a fraction, when multiplied by the original fraction, results in 1, effectively canceling out the coefficient. This step is vital for isolating y and revealing its value.

$\frac{2}{3} y = -6$

Multiply both sides by $\frac{3}{2}$:

$\frac{2}{3} y imes \frac{3}{2} = -6 imes \frac{3}{2}$

Simplify:

$y = -9$

This step is justified by the multiplication property of equality. Multiplying both sides by the reciprocal of the coefficient of y isolates the variable.

After arriving at a solution, it's essential to verify its correctness. This crucial step ensures that the calculated value of y indeed satisfies the original equation. Substituting the solution back into the original equation acts as a check, confirming that both sides of the equation remain equal. If the equation holds true after substitution, our solution is validated, providing confidence in our calculations. If the equation does not hold true, it indicates a potential error in the steps taken, prompting a review of the solution process. This verification step is not merely a formality; it's a safeguard against errors and a testament to the accuracy of our solution.

Substitute $y = -9$ into the original equation:

$\frac{2}{3} (-9) + 15 = 9$

Simplify:

$-6 + 15 = 9$

$9 = 9$

The solution is correct because it satisfies the original equation.

Solving equations is a step-by-step process, and each step is justified by fundamental properties of equality. These properties ensure that the equation remains balanced and that the solution obtained is correct. In the following sections, we provide a detailed justification for each step in solving the equation $\frac{2}{3} y + 15 = 9$.

Step 1: Subtracting 15 from both sides

The subtraction property of equality states that if we subtract the same number from both sides of an equation, the equation remains balanced. This property is based on the idea that an equation is like a balanced scale. If we remove the same weight from both sides of the scale, it will remain balanced. In our case, we subtract 15 from both sides to isolate the term with the variable y. This is a critical step in solving the equation because it moves us closer to isolating y on one side of the equation. By subtracting 15, we effectively cancel out the +15 on the left side, leaving us with a simpler equation to solve.

Step 2: Multiplying both sides by $\frac{3}{2}$

After subtracting 15, we have the equation $\frac{2}{3} y = -6$. To isolate y, we need to eliminate the coefficient $\frac{2}{3}$. The multiplication property of equality states that if we multiply both sides of an equation by the same non-zero number, the equation remains balanced. To eliminate the fraction, we multiply both sides by its reciprocal, which is $\frac{3}{2}$. The reciprocal of a fraction is obtained by swapping its numerator and denominator. When we multiply a fraction by its reciprocal, the result is 1, effectively canceling out the coefficient. This step is crucial because it isolates y and allows us to find its value. The multiplication property of equality is a powerful tool in algebra, enabling us to manipulate equations while preserving their balance.

Step 3: Verification

Verification is a critical step in solving equations. It ensures that the solution we have found is correct and satisfies the original equation. To verify the solution, we substitute the value of y back into the original equation and check if both sides of the equation are equal. If the equation holds true after substitution, our solution is correct. If the equation does not hold true, it indicates that there may be an error in our calculations. Verification provides us with confidence in our solution and helps us catch any mistakes we may have made. It is an essential step in the problem-solving process.

In conclusion, solving equations requires a systematic approach, with each step justified by fundamental properties of equality. By applying the subtraction and multiplication properties of equality, we successfully solved the equation $\frac{2}{3} y + 15 = 9$ and found that $y = -9$. Verification confirmed the correctness of our solution, ensuring that it satisfies the original equation. Understanding and applying these properties is essential for mastering algebraic equations and problem-solving in mathematics. The ability to solve equations is a fundamental skill that opens doors to more advanced mathematical concepts and applications. Practice and a solid understanding of these principles will empower you to tackle a wide range of mathematical challenges. Remember, each step in solving an equation has a purpose and a justification. By understanding these justifications, you gain a deeper appreciation for the elegance and logic of mathematics.