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Chapter 3: Problem 29

Solve each equation. $$-16-4 x=9$$

Short Answer

Expert verified
The solution is \(x = -\frac{25}{4}\).

Step by step solution

01

Isolate the Variable Term

The equation given is \(-16 - 4x = 9\). To isolate the variable term, start by adding 16 to both sides of the equation. This gives us \(-4x = 9 + 16\) which simplifies to \(-4x = 25\).
02

Solve for the Variable

The next step is to solve for \(x\) by dividing both sides of the equation by -4 to get: \[x = \frac{25}{-4}\] Simplifying gives \(x = -\frac{25}{4}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Solving Equations
Solving linear equations is about finding the value of the unknown variable that makes the equation true. In our case, the equation is \(-16 -4x = 9\). This is a typical linear equation, where the variable is a single unknown we want to find. The process of solving equations involves manipulating the equation in a series of lawful steps.

To solve, we transform the equation into simpler forms until we isolate the unknown. Each step requires performing an operation on both sides of the equation. This maintains the balance of the equation, much like balancing a see-saw.

Common operations include:
  • Adding or subtracting the same number from both sides of the equation.
  • Multiplying or dividing both sides by the same non-zero number.
By following these principles, one can systematically solve any algebraic equation.
Isolate the Variable
Isolating the variable is the cornerstone of solving any algebraic equation. It involves rearranging the equation to get the variable alone on one side. In the equation \(-16 - 4x = 9\), we want to isolate \(x\).

Here is the plan to isolate \(x\):
  • First, remove any constant terms from the side with the variable. This means we add 16 to both sides to eliminate the \(-16\) from the left. This transforms our equation to \(-4x = 25\).
  • Next, we divide each side by \(-4\) to leave \(x\) by itself. This step simplifies to \(x = -\frac{25}{4}\).
This two-step process of adding/subtracting first, then multiplying/dividing, is used to isolate the variable efficiently. Each step gets us closer to determining the variable's value.
Algebraic Expressions
An algebraic expression is a mathematical phrase that includes numbers, variables, and operations. In our exercise, the expression \(-16 - 4x\) combines constants and the variable \(x\).

Understanding expressions is crucial for solving equations because they form the building blocks of the problem. Here, the term \(-4x\) signifies the product of \(-4\) and \(x\), which is a key component in isolating the variable.

Expressions can be simplified to make equations easier to solve:
  • Combine like terms, if there are any.
  • Perform operations to reduce complexity and clarify the path to solving the equation.
Through simplification, expressions transform - opening the door to isolating variables and solving the equation accurately. Recognizing and reshaping algebraic expressions is fundamental in algebra.

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