Problem 62 - Algebra A Combined Function

Question

Solve each equation for y. See Section 2.5. $$ x-y=3 $$

Step-by-Step Solution

Answer

The solution is \( y = x - 3 \).

Step 1: Identify Terms to Isolate y

We begin with the given equation: \[ x - y = 3 \] Our goal is to solve for \( y \). To do this, we need to isolate \( y \) on one side of the equation.

Step 2: Move x to the Other Side

To isolate \( y \), we can move \( x \) to the other side of the equation by subtracting \( x \) from both sides to get:\[ -y = 3 - x \]

Step 3: Multiply by -1 to Solve for y

Since \( y \) has a negative coefficient, we need to multiply every term in the equation by \( -1 \) to solve for \( y \):\[ y = x - 3 \]

Step 4: Verify the Solution

To check our solution, substitute \( y = x - 3 \) back into the original equation to verify correctness: Plug \( y = x - 3 \) into \( x - y = 3 \) gives:\[ x - (x - 3) = 3 \]Simplifying this results in \( 3 = 3 \) which is correct. Hence, the solution is verified.

Key Concepts

Solving for Variables

When faced with any linear equation like \( x - y = 3 \), our objective is to solve for one of the variables, in this case, \( y \). The process of solving for a variable involves isolating it to one side of the equation. Here’s how to do it:

• Look at the equation and identify the variable you need to solve for. In this problem, that’s \( y \).
• Determine what other terms or variables are on the same side of the equation as \( y \). Here, \( x \) is on the left side together with \( y \).
• Move these other terms to the opposite side of the equation by performing operations such as addition, subtraction, multiplication, or division as needed.

By subtracting \( x \) from both sides, we effectively move \( x \) away from \( y \). This doesn't change the equality but helps in simplifying the expression to focus only on the variable of interest. Thus, we change \( x - y = 3 \) to \( -y = 3 - x \). From here, we focus on obtaining \( y \) with a positive coefficient. To continue solving, simply multiply all terms by \(-1\) to solve for \( y \), resulting in \( y = x - 3 \).

Equation Verification

Verification of your solution is essential in mathematics to ensure accuracy. Once you have solved for your variable, you should substitute your answer back into the original equation to verify it works. For this problem, once we've calculated \( y = x - 3 \), we substitute \( y \) back into the original equation \( x - y = 3 \). Doing this substitution would look like:

• Replace \( y \) with \( x - 3 \) in the original equation.
• This gives the equation \( x - (x - 3) = 3 \).
• Simplify the expression: \( x - x + 3 = 3 \).

After simplifying, you should find that both sides of your equation still agree – you should get \( 3 = 3 \), confirming the validity of your solution. Performing such checks helps reinforce confidence in your solution.

Algebraic Manipulation

Algebraic manipulation is a core skill in solving equations. It involves using arithmetic operations to rearrange and simplify equations. When manipulating the equation \( x - y = 3 \) to solve it:

• You subtract \( x \) from both sides, transforming the equation into \( -y = 3 - x \).
• Recognize that \( y \) has a negative coefficient (-1), thus multiplying every term by \(-1\) provides \( y = x - 3 \).

These manipulations allow us to rearrange the terms effectively, giving us a clear path to follow in solving for the desired variable. Key operations include:

• Adding or subtracting the same value from both sides of the equation.
• Multiplying or dividing each term by the same non-zero number.
• Understanding how different parts of an equation relate and change in response to these operations.

Mastering algebraic manipulation empowers you to approach linear equations and more complex algebraic expressions with confidence.