Problem 57 - Prealgebra

Question

Solve each equation. Check your solution. $$x-4=-6$$

Step-by-Step Solution

Answer

The solution is \(x = -2\).

Step 1: Isolate the Variable

The given equation is \(x - 4 = -6\). To isolate \(x\), we need to get rid of the \(-4\) that is being subtracted from \(x\). We can do this by adding \(4\) to both sides of the equation:\[x - 4 + 4 = -6 + 4\]This simplifies to:\[x = -2\]

Step 2: Verify the Solution

To ensure that our solution \(x = -2\) is correct, we substitute \(-2\) back into the original equation to check:\[(-2) - 4 = -6\]Simplifying the left side gives:\[-6 = -6\]Since both sides are equal, the solution \(x = -2\) is verified.

Key Concepts

Isolating the Variable

When solving equations, isolating the variable is a key step because it allows you to identify the value of the unknown. The goal is to have the variable by itself on one side of the equation.

To do this, we sometimes need to reverse operations affecting the variable. Imagine you're solving the equation \(x - 4 = -6\). The operation currently affecting \(x\) is subtraction (\(-4\)). To isolate \(x\), we add \(4\) to both sides of the equation:

• Add \(4\) to the left side: \(x - 4 + 4\)
• Add \(4\) to the right side: \(-6 + 4\)

This operation helps cancel out the negative \(4\) on the left, leaving \(x\) by itself:

• After simplification, you get \(x = -2\)

This step is crucial as it results in having a clear value for the variable, making it easy to interpret and check further.

Checking Solutions

Once the variable is isolated and you have an answer, checking the solution is vital. This confirms the solution is correct, ensuring it satisfies the original equation.

To check the solution for \(x = -2\), substitute this value back into the original equation \(x - 4 = -6\):

• Replace \(x\) with \(-2\): \((-2) - 4\)

Simplify to see if both sides of the equation are equal:

• Simplify: \(-2 - 4 = -6\)
• Comparison: \(-6 = -6\)

Since the left and right sides match, the solution \(x = -2\) is verified. This step isn’t just procedural; it reinforces the logic behind solving equations and ensures there’s no error in your simplified solution.

Integer Operations

Understanding integer operations is essential when isolating variables and checking solutions. Basic operations include addition, subtraction, multiplication, and division. Integer operations involve both positive and negative numbers, which are crucial when manipulating equations.

Consider the equation \(x - 4 = -6\). Here's how integer operations play a role:

• The equation shows subtraction. To cancel the \(-4\), you use its inverse operation, addition. Hence, add \(4\) to both sides.

When dealing with equations involving integers, keep in mind:

• Add and subtract negative numbers carefully. A negative plus a positive might reduce a number or even flip it positive.
• Consider the roles and interactions of numbers carefully. Missteps can lead to incorrect solutions.

Mastering these operations ensures accuracy when solving and verifying equations, essential skills in algebra.