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Chapter 2: Problem 6

State the property of equality you would use to solve each equation. Do not solve. $$-7 x=14$$

Short Answer

Expert verified
The Division Property of Equality would be used to solve this equation.

Step by step solution

01

Identify the Equation

The given equation is \( -7x = 14 \).
02

Determining the Property of Equality

In order to solve an equation where a variable is multiplied by a number, the division property of equality is typically used. This property states that if you divide both sides of an equation by the same nonzero number, the two sides remain equal.

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Key Concepts

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

Properties of Equality
The properties of equality are fundamental rules in algebra that help us manipulate equations without changing their truth. They ensure what we do to one side of the equation, we also do to the other, maintaining a balance. This balance is crucial for accurate calculations.
  • Addition Property of Equality: If you add the same number to both sides of an equation, the equality is still true. For example, if \(a = b\), then \(a + c = b + c\).
  • Subtraction Property of Equality: Subtracting the same number from both sides doesn't affect the truth of the equation, like \(a - c = b - c\).
  • Multiplication Property of Equality: If you multiply both sides by the same number, the relation holds, such as \(ac = bc\).
  • Division Property of Equality: This is the star of our example. If both sides are divided by the same non-zero number, the equation remains equal. For instance, \(\frac{a}{c} = \frac{b}{c}\), crucial for simplifying or solving equations.
These properties allow us to systematically solve equations while adhering to mathematical truth.
Solving Equations
Solving equations involves finding the value of the variable that makes the equation true. It is like solving a puzzle where each step brings you closer to the solution. In algebra, solving equations often involves inverse operations.

For the equation \(-7x = 14\), we use the division property of equality, as multiplying both sides by the reciprocal of \(-7\) helps in isolating \(x\).
  • Isolate the variable: We aim to have the variable on one side of the equation by itself.
  • Perform inverse operations: Since \(x\) is multiplied by \(-7\), we do the opposite 鈥� divide by \(-7\) to both sides. It results in \(x = \frac{14}{-7}\).
By keeping these steps in mind, you can tackle almost any basic algebraic equation.
Algebraic Concepts
Algebraic concepts extend beyond basic arithmetic and help us understand relationships between numbers and symbols. It's about structured thinking and finding solutions using abstract concepts.

In algebra, we work with variables, which are symbols, often letters, representing unknown values. They allow for general expressions rather than specific numbers, increasing flexibility of problem solving.
  • Variables and Constants: While variables can change, constants remain fixed. For example, in \(-7x = 14\), \(-7\) and \(14\) are constants.
  • Expressions and Equations: An expression is a combination of variables and constants without an equality sign, while an equation does include one. Equations set two expressions to be equal.
  • Balancing Equations: This concept is crucial, where operations apply equally to both sides of the equation, ensuring accurate solutions.
Understanding these algebraic concepts provides a strong foundation for solving complex mathematical problems.

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Use the subtraction property of equality to solve each equation. Check all solutions. $$a+9=-12$$
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