Problem 4 - Algebra A Combined Function

Question

Solve each equation. Check each solution. $$ y-9=1 $$

Step-by-Step Solution

Answer

The solution is \( y = 10 \). It satisfies the original equation.

Step 1: Isolate the variable

To solve the equation \( y - 9 = 1 \), we need to get \( y \) by itself on one side of the equation. To do this, add 9 to both sides of the equation. \[ y - 9 + 9 = 1 + 9 \] This simplifies the equation to \[ y = 10 \]

Step 2: Verify the solution

After finding that \( y = 10 \), we should check if this solution makes the original equation true. Substitute \( y = 10 \) back into the original equation \[ 10 - 9 = 1 \] Since both sides are equal, \( y = 10 \) is the correct solution.

Key Concepts

Solving Equations

Solving equations is a fundamental concept in algebra. It involves finding the value of a variable that makes an equation true. In simple terms, solving an equation means determining the unknown's value that when substituted into the equation, keeps both sides equal. For the equation \( y - 9 = 1 \), the goal is to find the value of \( y \) that satisfies this equality. The process usually involves manipulating the equation through arithmetic operations such as addition, subtraction, multiplication, or division. These operations help simplify the equation and solve for the unknown variable by isolating it on one side. Remember, the integrity of the equation must be maintained; thus any operation performed on one side must also be applied to the other.

Checking Solutions

Once you solve an equation, it is crucial to verify your solution. This step ensures the accuracy of your work. Verification involves substituting the found solution back into the original equation and checking if the equation holds true. For instance, after solving \( y - 9 = 1 \) and finding \( y = 10 \), we substitute \( 10 \) back into the original equation:

• Left side: \( 10 - 9 = 1 \)
• Right side: \( 1 \)

Both sides are indeed equal, confirming that \( y = 10 \) is correct. Verifying ensures that any potential errors during manipulation steps are caught, providing confidence in the solution's correctness. This habit is especially valuable when dealing with complex equations.

Isolation of Variable

Isolation of a variable is at the core of solving equations. It's the process of making one side of the equation contain only the variable you are solving for. To achieve this, all other numbers are moved to the opposite side of the equation. For \( y - 9 = 1 \), isolation is done by adding \( 9 \) to both sides, making the equation \( y = 10 \). This step-by-step strategy simplifies the equation drastically.Several key tips to remember when isolating variables include:

• Perform the same operation on both sides to maintain equality.
• Use inverse operations, like addition to undo subtraction.
• Simplify as much as possible at each step.

These techniques ensure that the process is both systematic and efficient, leading you directly to the solution.