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Chapter 2: Problem 84

Use the multiplication or division property of equality to solve each equation. $$-19 x=-57$$

Short Answer

Expert verified
The solution to the equation is \(x =3\).

Step by step solution

01

Identify the operation being applied to the variable

The variable \(x\) is being multiplied by \(-19\). The equation is \(-19x = -57\).
02

Apply the multiplication property of equality

To isolate the variable \(x\), divide both sides of the equation by \(-19\). This gives \(x = -57 / -19\).
03

Calculate the value of x

Upon calculating, the solution to the equation is \(x = 3\) as \(-57 / -19 = 3\).

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Key Concepts

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

Multiplication Property of Equality
In algebra, understanding the multiplication property of equality is essential for solving equations. This property states that if you have two equal quantities, you can multiply each side of the equation by the same nonzero number, and the equality will still hold true. Here's why it's useful:
  • It allows you to maintain equality while working toward isolating the variable you're trying to solve for.
  • You can use it to eliminate fractions and simplify complex equations.
  • It's a fundamental tool that supports solving a variety of algebraic problems.
For instance, if you have an equation such as \[-19x = -57\], knowing that multiplication is the operation affecting the variable leads us to use the inverse operation to simplify it. This principle assures that multiplying or dividing both sides does not change the equation's truthfulness.
Division Property of Equality
Similar to its multiplication counterpart, the division property of equality is a fundamental aspect of solving equations in algebra. This property suggests that you can divide both sides of an equation by the same nonzero number, and the equation remains balanced. This is because division is essentially the inverse of multiplication.
  • It helps in isolating the variable, a crucial step in solving for it.
  • Dividing both sides by the same number helps in simplifying the equation.
  • It is particularly useful when a variable is entangled with a coefficient that needs removing.
In the original exercise, the division property is applied to tackle the equation \[-19x = -57\]. By dividing each side by \(-19\), the equation simplifies to \[x = 3\]. This leaves us with the solution, effectively solving the equation.
Solving Equations
Solving equations is at the heart of algebra. It involves finding the value of the variable that makes the equation true. Here's a structured approach to solving equations:
  • First, identify the operation being applied to the variable.
  • Decide whether you need to use multiplication or division to cancel out the coefficient opposite of the variable.
  • Apply the appropriate property of equality - multiplication or division - to isolate the variable.
  • Calculate or simplify the resulting expression to find the variable value.
Taking the equation from our exercise \[-19x = -57\], the process involves recognizing that x is multiplied by -19. By dividing both sides of the equation by -19, we isolate \(x\). This approach leads us to the solution \(x = 3\), demonstrating the straightforward yet powerful nature of algebraic problem-solving.

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