Problem 10 - Elementary Algebra Within Reach

Question

Decide which operation you would use first to solve the equation. $$2 d+10=0$$

Step-by-Step Solution

Answer

The operation to be used first to solve the equation \(2d+10=0\) is subtraction.

Step 1: Subtract the constant on both sides

Subtract 10 from both sides of the equation to isolate the term with the variable \(d\). Doing this yields: \(2d + 10 - 10 = 0 - 10\).

Step 2: Simplify both sides of the equation

Simplify the left side of the equation by cancelling out the positive 10 and the negative 10, and the right side of equation by doing the subtraction operation. This gives: \(2d = -10\).

Step 3: Divide both sides by variable's coefficient

To completely isolate the variable \(d\), divide both sides of the equation by 2: \(2d/2 = -10/2\).

Step 4: Simplify to find the solution

Simplify the equation to find the solution. Doing these operations yields the solution: \(d = -5\).

Key Concepts

Operations with Equations

When solving algebraic equations, the order of operations is crucial. This process ensures that we handle each component of the equation correctly. Consider the equation "\(2d + 10 = 0\)". The first step involves identifying which operations are being performed in the equation:

• In the left side, there is addition and multiplication present: "addition by 10" and "multiplication by 2" with the variable \(d\).
• The goal is to undo these operations in a strategic manner.

Generally, we work from the outermost operation to the innermost. This means that we may start by reversing addition or subtraction before dealing with multiplication or division. Here, subtracting 10 is the first logical operation to perform. This is because it directly counteracts the addition of 10 in the equation. By subtracting 10 from both sides, we maintain the balance of the equation and start the process of isolating our variable \(d\).

Isolating Variables

Isolating variables involves getting the variable of interest alone on one side of the equation. This is essential to solve for the variable's value effectively. Here, our main objective is to isolate \(d\) in the equation "\(2d + 10 = 0\)".After subtracting 10 from both sides of the equation, we get "\(2d = -10\)". At this stage, \(d\) is still not isolated because it is part of the term "2d". The multiplication by 2 is the last operation that needs to be undone. We address this by dividing both sides of the equation by 2. As division is the inverse of multiplication, it successfully isolates \(d\):

• The left side becomes: \(2d/2 = d\).
• The right side becomes: \(-10/2 = -5\).

Thus, when we isolate \(d\), we discover that \(d = -5\). Isolating variables is like peeling away layers, one step at a time.

Algebraic Simplification

Algebraic simplification is the process of making an equation as straightforward as possible. Simplification means reducing expressions to their simplest form. Let's simplify the equation "\(2d = -10\)" to find \(d\). Here, simplification involves performing straightforward arithmetic operations like:

• Cancel addition or subtraction with their opposites, as we did with "\(+10\) and \(-10\)".
• Use division to eliminate the coefficient from the variable term, which we did by dividing "2" on both sides.

By stripping away extra terms and coefficients, simplification helps us express equations clearly. It ensures our solution – here, \(d = -5\) – is neatly packaged without extra clutter. Simplification provides clarity, making the solutions easily interpretable for continued use in problem-solving.