Problem 73 - Prealgebra

Question

The problems below review some of the material on solving equations. Reviewing these problems will help you with the next section. Solve. $$x+3=-6$$

Step-by-Step Solution

Answer

x = -9

Step 1: Identify the Equation

The given equation to solve is \( x + 3 = -6 \). Our goal is to isolate \( x \) on one side of the equation to find its value.

Step 2: Subtract 3 from Both Sides

To isolate \( x \), we need to remove the +3 from the left side. Subtract 3 from both sides of the equation:\[(x + 3) - 3 = -6 - 3\]This simplifies to:\[x = -9\]

Step 3: Verify the Solution

To verify the solution, substitute \( x = -9 \) back into the original equation:\[(-9 + 3) = -6\]Simplifying gives \( -6 = -6 \), which is true, confirming that our solution is correct.

Key Concepts

Isolating Variables

In solving linear equations, one of the key skills to master is isolating the variable. This means we rearrange the equation so that the variable, usually represented by a letter like \( x \), stands alone on one side of the equation.
When we have an equation like \( x + 3 = -6 \), our task is to determine what number \( x \) represents. To do this, we need to 'undo' what is being done to \( x \). In our example, \( x \) is being increased by 3. So, we perform the opposite operation to eliminate this constant:

• Subtract 3 from both sides of the equation. This keeps the equation balanced while removing the constant from \( x \).
• The resulting equation becomes: \[(x + 3) - 3 = -6 - 3\]
• This simplifies to: \[x = -9\]

By isolating \( x \), we find that \( x = -9 \). The process of isolating the variable is fundamental in solving linear equations as it helps identify the value of the variable.

Verifying Solutions

After isolating the variable and obtaining a solution, it's crucial to verify that the solution is correct. Verifying solutions ensures that no mistakes were made during the process, and that the solution indeed satisfies the original equation.
Verification requires substituting the found solution back into the original equation. For instance, if we have solved \( x + 3 = -6 \) and found \( x = -9 \):

• Substitute \( -9 \) for \( x \) in the original equation.
• Re-evaluating gives us: \[(-9 + 3) = -6\]
• Simplifying, we see: \[-6 = -6\]

Both sides of the equation are equal, confirming our solution is accurate. Verifying solutions is a simple yet essential step to ensure the integrity of our results when solving equations.

Equations Review

Reviewing equations is about understanding how different operations affect the equality and help find solutions. Equations are mathematical statements that assert the equality of two expressions, and the goal is usually to find the value of unknown variables.
In linear equations, like \( x + 3 = -6 \), we often:

• Identify the type of operation affecting the variable (e.g., addition, subtraction).
• Apply inverse operations to both sides to maintain the balance of the equation.
• Isolate the variable to determine its true value.

By practicing problems and reviewing each step carefully, solving equations becomes more intuitive. Equations are everywhere in mathematics, forming the basis of algebraic reasoning and problem-solving.
Thus, regularly revisiting and reviewing the steps provides a solid foundation in understanding how to manipulate and solve them effectively.