Problem 60 - Elementary Algebra

Question

Solve. $$ -916 x+43=43 $$

Step-by-Step Solution

Answer

The solution is \(x = 0\).

Step 1: Isolate the variable term

To isolate the term containing the variable \(x\), subtract 43 from both sides of the equation: \[-916x + 43 - 43 = 43 - 43 \]which simplifies to \[-916x = 0\].

Step 2: Solve for the variable

Divide both sides of the equation by -916 to solve for \(x\): \[x = \frac{0}{-916}\].Since dividing zero by any non-zero number is zero, we have\[x = 0\].

Key Concepts

Variable Isolation

Variable isolation is a fundamental step in solving algebra equations. The goal is to have the variable, usually represented as 'x', standing alone on one side of the equation. This makes it easier to determine its value.
A typical approach involves performing operations that "undo" what has been done to the variable. For example:

• If something is added to the variable, you subtract it from both sides of the equation.
• If something is subtracted, you add it.
• Similarly, if a variable is multiplied, you divide, and vice versa.

This is just like peeling layers of an onion to get to the core. In our exercise, we started by subtracting 43 from both sides to isolate the term \(-916x\). This is the first step before you can actually solve for \(x\).
Remember, whatever operation you do to one side of the equation, you must do to the other to maintain balance.

Solving Linear Equations

Linear equations are equations of the first degree, meaning they involve no exponents greater than one. These equations often involve operations to isolate the variable. In the exercise given, we had a basic linear equation with a single variable, \(-916x + 43 = 43\).
The process of solving begins once we've isolated the variable term. It's about systematically simplifying the equation until only 'x' remains on one side.
This often involves a combination of addition, subtraction, multiplication, and division to reach the variable's value. It is crucial to keep the equation balanced throughout by performing the same operations on both sides.

Division in Equations

The division operation is common when solving equations, particularly when you have isolated a term with a coefficient in front of the variable. To get "x" by itself, you need to undo this multiplication.
In our exercise, once we had the equation \(-916x = 0\), dividing both sides by -916 allows us to solve for \(x\).
This technique is used because division is the inverse of multiplication, effectively canceling out the coefficient.

• Remember, dividing zero by any non-zero number will always result in zero.
• Thus, when \(0\) was divided by \(-916\), \(x\) became \(0\).

Understanding this basic property is essential for solving equations that result in zero on one side. This is a critical final step, confirming the solution once the variable is isolated.