Problem 19 - Prealgebra

Question

Use the multiplication property of equality to solve each of the following equations. In each case, show all the steps. $$-7 x=-42$$

Step-by-Step Solution

Answer

\(x = 6\)

Step 1: Identify the Coefficient of the Variable

The equation given is \(-7x = -42\). Here, the coefficient of the variable \(x\) is \(-7\). Our goal is to solve for \(x\) by isolating it, which requires us to get rid of this coefficient.

Step 2: Apply the Multiplication Property of Equality

According to the multiplication property of equality, both sides of the equation may be multiplied (or divided) by the same nonzero number without changing the equation's solution. Divide both sides by \(-7\) to isolate \(x\):\[x = \frac{-42}{-7}\].

Step 3: Simplify the Right Side of the Equation

Calculate \(\frac{-42}{-7}\). Since dividing two negative numbers results in a positive number:\[x = 6\].

Key Concepts

Multiplication Property of Equality

In mathematics, the multiplication property of equality is a fundamental principle that allows us to maintain the balance and truth of an equation while isolating variables. This property states that multiplying (or dividing) both sides of an equation by the same nonzero number does not change the equality of the equation.
If you have an equation like \(-7x = -42\), you can use this property to solve for the unknown variable, which in this case is \(x\).

• Identify the coefficient: This is the number directly attached to the variable. In our example, it is \(-7\).
• Divide both sides of the equation by the coefficient: This means you will divide by \(-7\) on both sides, as shown in the solution: \(x = \frac{-42}{-7}\).

This simple process isolates the variable, allowing you to find its value without altering the truth of the original equation. It’s crucial that the number you choose to multiply or divide by is not zero, as dividing by zero is undefined.

Solving Linear Equations

Linear equations are equations that represent straight lines when graphed, characterized by constants and a single variable raised to the power of one. A standard form of a linear equation can be seen in expressions like \(ax + b = c\), where \(a\), \(b\), and \(c\) are constants.
To solve linear equations, the objective is to isolate the variable on one side of the equation:

• First, identify terms involving the variable and those that are constants.
• Use inverse operations, like addition and subtraction or multiplication and division, to move terms and isolate the variable.
• Perform the same operation on both sides to keep the equation balanced.

In our example, \(-7x = -42\), the return to balance involves dividing both sides by \(-7\) to solve for \(x\). As you work through these steps, remember that the manipulation of the equation must maintain equality at each stage.

Integer Operations

Integer operations are basic arithmetic functions performed on whole numbers, which include positive numbers, negative numbers, and zero. Understanding these is crucial in simplifying and solving equations effectively.

• **Addition and Subtraction**: Adding and subtracting integers involves looking at their signs. Two positive numbers or two negative numbers added together increase the original magnitude but the sign of the result will depend on the integers involved.
• **Multiplication and Division**: The product or quotient of two integers depends on the signs of those numbers. Multiplying or dividing two positive or two negative numbers results in a positive number.
• Conversely, multiplying or dividing integers with different signs yields a negative result.

In the given exercise, you simplify \(-42 \/ -7\) using division. Because both numbers are negative, their quotient is positive. Thus, \(x = 6\). Understanding how negative and positive integers interact can greatly help in identifying solutions more accurately and quickly.