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Chapter 2: Problem 36

Solve the given equation for \(\mathrm{x}\). \(\frac{x}{-3}=-7\)

Short Answer

Expert verified
x = 21.

Step by step solution

01

Understand the Equation

The given equation is \( \frac{x}{-3} = -7 \). It implies that when \( x \) is divided by -3, the result is -7. We need to find the value of \( x \) that satisfies this equation.
02

Eliminate the Fraction

To solve for \( x \), we need to eliminate the fraction. We do this by multiplying both sides of the equation by -3 to cancel out the denominator:\[ x = -7 \times (-3) \]
03

Simplify the Equation

Calculating the right side of the equation:\[ x = 21 \]This means that when \( x = 21 \), the original equation is satisfied.

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Key Concepts

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

Prealgebra
Prealgebra forms the foundation of more advanced mathematical concepts. It's your stepping stone to algebra and beyond. In prealgebra, we deal with basic arithmetic and the introduction of variables, like the letter \( x \) in our equation. Understanding how to manipulate these variables is crucial.
The goal is to learn how to perform operations that can simplify and solve equations. This starts with understanding basic operations like addition, subtraction, multiplication, and division, and then applying these operations to algebraic expressions.
  • Grasping the concept of a variable as an unknown quantity.
  • Learning to perform arithmetic operations on both sides of an equation.
  • Maintaining balance in an equation while solving for \( x \).
Once you've mastered prealgebra, you'll find it much easier to tackle more complex algebraic equations.
Fraction Elimination
Solving equations with fractions can seem tricky, but with practice, it becomes straightforward. The key is to eliminate the fraction early on, making your equation simpler to work with. In our example, \( \frac{x}{-3} = -7 \), the variable \( x \) is divided by \(-3\).
To eliminate the fraction:
  • Multiply both sides of the equation by the denominator \(-3\). This will cancel out the fraction.
  • The equation then becomes \( x = -7 \times (-3) \).
This process transforms the initial equation into a simpler form without fractions, making it easier to find the solution for \( x \). It's important to apply this multiplication to both sides to maintain balance and not change the equality of the equation.
Multiplication of Negatives
Multiplying negative numbers can be confusing at first, but there are simple rules to follow that make it easier. When you multiply two negative numbers, the product is positive. This might seem counterintuitive, but it's a key rule to remember.
In the equation \( x = -7 \times (-3) \), you're multiplying two negative numbers:
  • \(-7\) is negative.
  • \(-3\) is also negative.
  • When multiplied, \(-7 \times (-3) = 21\). The negatives cancel out to give a positive product.
This rule helps in simplifying equations and is crucial when solving for variables. Consistently applying this rule helps avoid mistakes and builds a strong mathematical foundation. Recognizing that two negatives make a positive is key to successfully solving many algebraic equations.

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