Problem 79 - Intermediate Algebra

Question

Solve each equation. $$ -x+12=-17 $$

Step-by-Step Solution

Answer

The solution to the equation is \(x = 29\).

Step 1: Eliminate the Negative Sign

The given equation is \(-x + 12 = -17\). The first step is to eliminate the negative sign from the \(-x\) term. Since \(x\) is being negated, multiply both sides of the equation by -1 to get:\(x - 12 = 17\).

Step 2: Isolate the Variable

Now that we have \(x - 12 = 17\),we need to solve for \(x\). Add 12 to both sides of the equation to isolate \(x\):\(x = 17 + 12\).

Step 3: Simplify the Expression

Simplify the right side of the equation:\(x = 29\).

Key Concepts

Solving Equations

Solving equations is like solving a puzzle! Equations are mathematical statements that show the equality of two expressions. To solve them, you want to find the value of the variable that makes the statement true. It's like asking, "What number represents this unknown in the equation?" Let's explore how you can do that:- Understand the equation given: It usually consists of two sides, divided by an equal sign. For example, \(-x + 12 = -17\).- Your goal is to manipulate the equation so that the variable (typically \(x\)) is alone on one side of the equation.- Avoid making any mistakes by doing the same operation on both sides of the equation. This keeps the balance, just like balancing a scale.Breaking down the problem step by step helps you see the solution. Every move you make in solving keeps everything fair and true, which is essential to solving equations correctly.

Isolating Variables

Isolating the variable is a critical step in solving equations. Imagine it as clearing the path for the variable to stand alone. Once the variable is by itself on one side of the equation, you’ll know its value. Here's how you can successfully isolate variables:- Look at the given equation, say: \(x - 12 = 17\).- Identify parts being added or subtracted from \(x\). In our example, 12 is subtracted.- To isolate \(x\), perform the opposite operation. If 12 is subtracted, then add 12 to both sides to maintain the balance: \(x - 12 + 12 = 17 + 12\). - This will result in \(x = 29\).Step-by-step operations such as adding or subtracting help you to isolate \(x\). Remember, whatever you do to one side, you must do to the other!

Negative Sign Elimination

Handling negative signs in equations can trip some students up. However, eliminating a negative sign is crucial for clarity and progress in solving an equation. Here's a simple way to eliminate negative signs:- Start with an equation like \(-x + 12 = -17\).- The \(-x\) can be confusing, so eliminate the negative by multiplying every term in the equation by -1.- This transformation gives us a clearer equation: \(x - 12 = 17\).Multiplying by -1 swaps the signs, helping to simplify the equation. Once all the terms are in a straightforward form, you can work through solving it more easily. It's a little trick that makes a big difference when solving equations efficiently.