Problem 7 - Fundamentals of Mathematics

Question

$$a-4=11$$

Step-by-Step Solution

Answer

The solution is \(a = 15\).

Step 1: Identify the Equation Components

The given equation is \(a - 4 = 11\). Here, \(a\) is the variable we need to solve for. The number 4 is being subtracted from \(a\), and the equation equals 11.

Step 2: Isolate the Variable

To isolate \(a\), we need to eliminate the -4 from the left side of the equation. This can be done by adding 4 to both sides of the equation. The equation will then become \((a - 4) + 4 = 11 + 4\).

Step 3: Simplify Both Sides

After adding 4 to both sides, the left side simplifies to \(a\) and the right side becomes 15. Thus, the equation simplifies to \(a = 15\).

Step 4: Verify the Solution

Substitute \(a = 15\) back into the original equation to verify: \(15 - 4 = 11\). Since this is true, \(a = 15\) is the correct solution.

Key Concepts

Understanding Variables

In mathematics, variables are symbols used to represent unknown values. Common symbols include letters like \(x\), \(y\), or \(a\), as in our equation example, \(a - 4 = 11\). Using variables allows us to formulate and solve equations more easily.
They act as placeholders for numbers we need to find. In this context, solving equations means finding which specific number makes the equation true.
Variables are essential because they can handle different values, generalize mathematical expressions, and simplify complex problems. They allow us to form an equation which can be manipulated to solve for an unknown, making them a fundamental part of algebra.

Applying the Isolation Method

The isolation method is central to solving equations as it focuses on "isolating" the variable on one side of the equation. To solve for \(a\) in \(a - 4 = 11\), the goal is to have \(a\) by itself on the left side of the equation. Here is how we do it:

• Identify what is being added, subtracted, multiplied, or divided with the variable.
• Use inverse operations to remove these elements from the variable's side of the equation. In this example, since \(4\) is subtracted from \(a\), we add \(4\) to both sides.

This process is essential because ensuring the variable is isolated helps clarify the mathematical relationship between the variable and the constants of the equation.
By executing these steps correctly, you balance the equation and solve for the unknown variable.

Verification of Solution

After finding a solution, it's crucial to verify it, ensuring that the number satisfies the original equation. In this exercise, we found \(a = 15\) by isolating \(a\). Verification involves substituting this value back into the original equation.
For our equation, substitute \(15\) for \(a\): \(15 - 4 = 11\).

• If both sides of the equation are equal after the substitution, as they are here (11 = 11), the solution is verified as correct.
• If they are not equal, re-evaluate the steps to check for any mistakes.

Verification provides confidence that the solution is accurate, prevents errors, and reinforces an understanding of the relationship between the equation's elements.