Problem 55 - Algebra 1: Concepts and Skills

Question

SOLVING EQUATIONS Solve the equation. $$ a-3=-2 $$

Step-by-Step Solution

Answer

The solution to the equation \( a - 3 = -2 \) is \( a = 1 \).

Step 1: Identify the equation

The given equation is \( a - 3 = -2 \). Here, 'a' is the variable that needs to be determined.

Step 2: Rearranging the equation

The goal is to isolate 'a'. To do this, add '3' to both sides of the equation. This gives \( a = -2 + 3 \), as the '3' on the left side and the '3' on the right side cancel each other out.

Step 3: Simplifying the right hand side

Simplify the right hand side of the equation to find the value of 'a'. The right hand side simplifies to \( a = 1 \).

Key Concepts

Isolate the variable

One of the fundamental steps in solving any equation is isolating the variable. This means rearranging the equation so that the unknown variable stands alone on one side, usually the left. By doing this, you get a clearer picture of what the variable actually equals. In the exercise given, the aim was to get 'a' by itself.

• We had the equation: \( a - 3 = -2 \).
• To isolate 'a', we needed to eliminate any numbers on the same side as 'a'. This involved some math magic which we'll talk about below!
• This magic is really just reversing operations to undo the math around 'a' until it's alone on its side of the equals sign.

With 'a' isolated, we can easily see what it's equal to directly.

Equation simplification

Equation simplification is the process of making an equation easier to work with, often a key step right after isolating the variable. In our example, simplifying helped confirm the value of 'a'.

• We started with: \( a = -2 + 3 \).
• By simplifying, we mean performing actual arithmetic calculation on this expression to get a clear answer.
• For this example, when you calculate \( -2 + 3 \), you find that \( a = 1 \).

The simplification gives the solution in the simplest form possible, making it very clear and precise.

Additive inverse

To isolate variables, we often use the concept of the additive inverse. This is a crucial step because it helps remove numbers from the variable's side of the equation. The additive inverse for a number is what you add to the number to make it zero. That’s how you 'cancel out' numbers!

• In our original equation, \( a - 3 = -2 \), the additive inverse of \(-3\) is \(3\).
• Adding \(3\) to both sides helps in neutralizing the \(-3\) next to 'a', which leads to: \( a = -2 + 3 \).
• Using the additive inverse keeps the equation balanced since whatever you do to one side, you must do to the other.

This balancing act with additive inverses is key in ensuring you solve equations correctly without altering their meaning.