Problem 34 - Prealgebra

Question

Solve each equation. Check your solution. $$10=-9-x$$

Step-by-Step Solution

Answer

The solution is \( x = -19 \).

Step 1: Identify the Equation

We are given the equation to solve: \( 10 = -9 - x \). The goal is to isolate \( x \) on one side of the equation to find its value.

Step 2: Add 9 to Both Sides

To isolate \( x \), we first add 9 to both sides of the equation to eliminate the \(-9\) from the right side. This gives us\[ 10 + 9 = -9 - x + 9 \] which simplifies to \[ 19 = -x \].

Step 3: Multiply by -1

Now, to get \( x \) by itself, we multiply both sides of the equation by \(-1\) to change the sign of \( x \). Therefore, \[ -1 imes 19 = -1 imes -x \] which gives \[ -19 = x \].

Step 4: Verify the Solution

Replace \( x \) in the original equation with \(-19\) to check the solution: \[ 10 = -9 - (-19) \]. Calculate the right side as \(-9 + 19 = 10\), which matches the left side. Hence, the solution \( x = -19 \) is correct.

Key Concepts

Solving Equations

When you first see an equation, the main goal is to find the value of the unknown quantity, often represented by a variable like \( x \). In simple terms, solving an equation means finding numbers that make the equation true. Equations can vary in complexity, but the basic idea remains the same.

• Start by looking at the equation carefully. Here, you identify what you have and what you need to do to find the solution.
• The objective is to isolate the variable (e.g., \( x \)) on one side of the equality sign to discover its value.
• We often employ opposite operations such as addition versus subtraction or multiplication versus division to achieve this.

In the example of the equation \( 10 = -9 - x \), our focus was to get \( x \) by itself so we can know what \( x \) equals. This is where a structured approach in solving comes in handy, leading us to our next important step.

Isolation of Variables

The core aim of isolation is to move all terms involving the variable to one side of the equation. This helps us to express the variable as a standalone term that reveals its actual value.To isolate the variable, follow these simple steps:

• Identify the term that contains the variable you want to isolate.
• Use inverse operations to remove other terms from the side of the equation where the variable is. For instance, if there's a subtraction affecting the variable, add the same number on both sides.
• Simplify the equation continuously after each operation.

In our example, adding 9 to both sides of \( 10 = -9 - x \) effectively cancels out the \(-9\) on the right and simplifies the equation to \( 19 = -x \). Multiplying both sides by \(-1\) further isolates \( x \), helping us find \( x = -19 \). By organizing our steps as such, we can clearly see the pathway to isolating and determining the value of the variable.

Checking Solutions

Once we've found a solution, like \( x = -19 \) in the original equation, it's crucial to verify that this solution is correct. This validation process is known as checking solutions.Checking helps us ensure accuracy and understanding:

• Substitute the value you found back into the original equation.
• Perform all calculations as per the equation to see if both sides match.
• If both sides are equal upon substitution, your solution is verified. If not, recheck your steps for any possible errors.

For example, substituting \( x = -19 \) back into \( 10 = -9 - x \), we check by calculating \(-9 - (-19)\) which simplifies to \( 10 \). Since both sides of the equation equal 10, this confirms our solution is indeed correct. Checking solutions is a vital step that not only proves the correctness of your work but also boosts your confidence in handling similar problems in the future.