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Chapter 3: Problem 54

Solve each equation. \(9-14=\frac{x}{-12}\)

Short Answer

Expert verified
The solution is \( x = 60 \).

Step by step solution

01

Simplify the Left Side

First, we simplify the left side of the equation. Calculate the difference: \[ 9 - 14 = -5 \] Thus, the equation becomes: \[ -5 = \frac{x}{-12} \]
02

Isolate the Variable

To solve for \( x \), we need to isolate it. Since \( x \) is divided by \(-12\), we multiply both sides of the equation by \(-12\) to cancel out the denominator: \[ -5 \times -12 = x \]
03

Calculate the Value of x

Calculate the product on the left side: \[ -5 \times -12 = 60 \] So, we have: \[ x = 60 \]

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Key Concepts

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

Linear Equations
Linear equations are fundamental components of algebra. They typically involve variables raised to the power of one. In other words, they are equations of the first degree. These equations take the form \( ax + b = c \). The main characteristic of linear equations is their simplicity and predictability. They graph as straight lines when plotted on a coordinate plane. A linear equation can have one, none, or infinitely many solutions, depending on the number of variables and constraints set by the equation itself.
  • Linear equations can be solved using basic algebraic operations.
  • They often represent real-world scenarios, such as financial calculations or physics problems.
Understanding linear equations helps in tackling more complex algebra problems.
Isolation of Variables
Isolation of variables is a critical strategy in solving equations, especially in algebra. This process involves manipulating the equation to get the variable by itself on one side of the equation. When dealing with the equation \(-5 = \frac{x}{-12}\), our goal is to isolate \(x\).
  • Begin by determining the operation affecting the variable鈥攊n this example, division by \(-12\).
  • To isolate, perform the inverse operation on both sides of the equation; here, multiplying both sides by \(-12\).
This method enables us to find the specific value of the variable that satisfies the equation. Remembering to apply the same operation to both sides keeps the equation balanced, which is key in maintaining the correctness of the solution.
Simplification in Algebra
Simplification in algebra involves reducing an equation or expression to its simplest form. This often makes it easier to solve. In the equation \(9 - 14 = \frac{x}{-12}\), the left side simplifies to \(-5\). Simplifying is the first step you should take when tackling equations since it can streamline the solution process considerably.
  • Identify like terms and combine them.
  • Perform arithmetic operations where applicable.
Simplifying can also involve factoring expressions, canceling common terms, or distributing values. Mastering simplification allows for clearer understanding and faster problem-solving, as it eliminates unnecessary complexity in algebraic tasks.

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