Problem 14 - Prealgebra

Question

Solve each equation. Check your solution. $$y+7.2=21.9$$

Step-by-Step Solution

Answer

The solution is \( y = 14.7 \).

Step 1: Isolate the Variable

To solve for \( y \), begin by isolating \( y \) on one side of the equation. The original equation is:\( y + 7.2 = 21.9 \)Subtract 7.2 from both sides to get:\( y = 21.9 - 7.2 \)

Step 2: Perform the Calculation

Now, subtract 7.2 from 21.9 to find the value of \( y \):\( y = 14.7 \)

Step 3: Verify the Solution

To ensure the solution is correct, substitute \( y = 14.7 \) back into the original equation:\( 14.7 + 7.2 = 21.9 \)Calculate the left side:\( 21.9 = 21.9 \)Since both sides of the equation are equal, \( y = 14.7 \) is indeed the correct solution.

Key Concepts

Algebraic Equations

Algebraic equations form the backbone of algebra, allowing us to find unknown values within logical mathematical statements. An algebraic equation includes variables, numbers, and operations. The "+" and "=" symbols often feature prominently, as seen in equations like \( y + 7.2 = 21.9 \). Here, \( y \) is a variable, an unknown number we must solve for, and the rest are known values. Understanding algebraic equations involves recognizing how to manipulate these components to find the value of the variable.
We apply operations with constants (such as numbers) in a sequence that maintains the balance of the equation. If you make a change to one side, you must make a corresponding change to the other to keep the equation balanced. By mastering these principles, algebra becomes a powerful tool for solving real-world and theoretical problems.

Isolate Variable

To solve an algebraic equation for a particular variable, you need to isolate that variable on one side of the equation. This process often involves undoing or reversing operations applied to the variable. In our example, the equation is \( y + 7.2 = 21.9 \). To isolate \( y \), you perform the inverse of adding 7.2, which is subtracting 7.2.
Follow these steps to isolate \( y \):

• Recognize that \( y \) is combined with 7.2 through addition.
• Subtract 7.2 from both sides of the equation: \( y + 7.2 - 7.2 = 21.9 - 7.2 \).
• Consequently, this simplifies to \( y = 14.7 \).

Learning to isolate variables is a critical skill because it unlocks your ability to solve for unknowns in more complex equations. This foundational skill is commonly employed to make equations easier to manage and solve.

Check Solution

Verifying the solution of an equation is a crucial step to confirm its accuracy. Once you have isolated the variable and found its value, substitute it back into the original equation to ensure both sides are equal. This process is known as checking or verifying the solution, and it ensures your calculations haven't led to an error.
For example, substituting \( y = 14.7 \) back into the original equation \( y + 7.2 = 21.9 \) results in:

• Left side: \( 14.7 + 7.2 = 21.9 \)
• Right side: \( 21.9 \)
• Since \( 21.9 = 21.9 \), the solution is verified.

Checking your work is a smart habit that reinforces confidence in your solution and deepens your understanding. It shows that you not only completed the steps but also understood the process.