Problem 85 - Algebra 1

Question

Solve the equation. Check the result. (Review 3.4 ) $$y-9=2$$

Step-by-Step Solution

Answer

The solution for the equation \(y - 9 = 2\) is \(y = 11\).

Step 1: Analyze the Equation

Analyzing the equation shows that we have \(y - 9 = 2\). Here, we are trying to solve for \(y\). Thus, we need to isolate \(y\) on one side of the equals sign. We do this by adding '9' to both sides.

Step 2: Add 9 to Both Sides

When you add 9 to both sides of the equation, the left side becomes \(y\). This is because \(-9 + 9 = 0\), and thus it cancels out on the left. On the right side, you simply add \(2 + 9\), which equals 11. The equation then becomes \(y = 11\).

Step 3: Check the Result

To ensure our solution is correct, we plug our value for \(y\) back into the original equation to see if it holds true. Substitute \(y\) with 11 in the original equation, we get: \(11 - 9 = 2\), which simplifies to \(2 = 2\). This equation is true, which confirms that our solution is correct.

Key Concepts

Understanding Linear Equations

Linear equations are foundational in algebra. They are like simple puzzles where you are given a mathematical statement with unknown variables. When you solve a linear equation, your goal is to find the value of the variable that makes the equation true.
A linear equation can typically be written in the form of \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants, and \(x\) is the variable we want to solve for. The equation given in the exercise is \(y - 9 = 2\). Here, \(y\) is the variable, and we want to find the number that, when subtracted by 9, gives 2.
When approaching linear equations, it’s helpful to remember that your job is to isolate the variable. Once the variable is isolated, you find the solution. This involves performing algebraic manipulations, which may include adding, subtracting, multiplying, or dividing both sides of the equation by the same number. By doing so, you maintain the balance of the equation, much like a scale.

• Step 1 is always to identify your goal: isolate the variable.
• Step 2 involves carrying out the appropriate operations to achieve this goal.

Checking Solutions for Accuracy

Once you have found a solution to a linear equation, it's crucial to check your work. This helps ensure no errors were made along the way.
Checking a solution means substituting the value you found for the variable back into the original equation. If both sides of the equation are equal, then your solution is correct. In our example, we found \(y = 11\). To check it, we replace \(y\) in the original equation, \(y - 9 = 2\), with 11 to see if it holds true.

• Substitute: \(11 - 9\)
• Calculate: \(11 - 9 = 2\)

Since we end up with \(2 = 2\), the solution checks out. This process might seem simple, but it’s an important step to verify the accuracy of your solution. Without checking, it’s easy to overlook simple errors or missteps made during solving.
Employing this double-check method will increase your confidence in your answers and improve your problem-solving skills in the long run.

The Art of Algebraic Manipulation

Algebraic manipulation is the method of rearranging and simplifying equations to isolate a variable. It's like arranging pieces on a chessboard to achieve a certain configuration. This skill is not only critical for solving algebra problems but also forms a basis for higher math and applied fields.
In the given example, the phrase "manipulate the equation" means adding 9 to both sides in the equation, \(y - 9 = 2\). Why do we add 9? Because the inverse of subtraction is addition, and we want to cancel out \(-9\) so that \(y\) stands alone on one side of the equation. As such:

• Add 9 to both sides: \(y - 9 + 9 = 2 + 9\)
• This simplifies to: \(y = 11\)

This manipulation process relies on understanding operations and their inverses. Whether you are adding, subtracting, multiplying, or dividing, always apply these operations equally to both sides of the equation. This keeps the mathematical relationship balanced and leads to a correct solution. In time, you'll discover that strong algebraic manipulation skills are not only practical in solving equations but also indispensable in many areas of math and science.