Problem 13 - Prealgebra

Question

Solve each equation. $$x-4=7$$

Step-by-Step Solution

Answer

The solution is \(x = 11\).

Step 1: Understand the Equation

The original equation is given as \(x - 4 = 7\). Our goal is to solve for \(x\) that makes this equation true.

Step 2: Isolate the Variable

To solve the equation \(x - 4 = 7\), add 4 to both sides of the equation to isolate \(x\). This gives us:\[x - 4 + 4 = 7 + 4\]

Step 3: Simplify the Equation

Simplify both sides of the equation:\[x = 11\]

Step 4: Verify the Solution

Substitute \(x = 11\) back into the original equation to verify:\[11 - 4 = 7\]Since both sides of the equation are equal, \(x = 11\) is the correct solution.

Key Concepts

Solving Equations

Solving equations is a fundamental skill in algebra and involves finding the value of a variable that makes an equation true. Imagine an equation like a balance scale, and our goal is to keep it balanced by performing the same operation on both sides. For an equation like \(x - 4 = 7\), we want to find out what number \(x\) would be to make the two sides equal.

To solve such equations, follow these steps:

• Identify the variable you need to solve for, which in this case is \(x\).
• Consider what operations are applied to the variable and think about reversing them to solve for the variable.
• Perform the same operation on both sides of the equation to maintain balance and solve for the variable.

These steps help simplify the equation and isolate the variable, paving the way for finding the solution.

Isolation of Variables

Isolating the variable is key in solving equations. The idea is to get the variable by itself on one side of the equation. This process often involves undoing operations applied to the variable, like addition, subtraction, multiplication, or division.

For instance, in the equation \(x - 4 = 7\), the goal is to isolate \(x\). We reverse subtracting 4 by adding 4 to both sides of the equation:

• This gives us \(x - 4 + 4 = 7 + 4\).
• When simplified, it results in \(x = 11\).

Remember, whatever operation you do to one side of the equation, you must do to the other. This keeps the equation in balance, much like a balanced scale.

Equation Verification

Once you've found a solution, it's crucial to verify your work. Verification ensures that your solution is correct and satisfies the original equation. This step acts as a double-check for your computations.

To verify the solution of \(x = 11\) for the equation \(x - 4 = 7\), follow these steps:

• Substitute the value \(x = 11\) back into the original equation.
• Calculate to see if both sides of the equation remain equal: \(11 - 4\).
• If correct, the result should be \(7\), confirming that \(11 - 4 = 7\).

Verification confirms that you have the correct solution and reinforces your understanding of the problem-solving process.