Problem 24 - Elementary Algebra

Question

Solve. $$ -9 x=63 $$

Step-by-Step Solution

Answer

x = -7

Step 1: Identify the Equation

The given equation is \(-9x = 63\). Our goal is to find the value of \(x\).

Step 2: Isolate the Variable

To isolate \(x\), we need to divide both sides of the equation by \(-9\). This operation will help us find the value of \(x\). The equation becomes \(x = \frac{63}{-9}\).

Step 3: Simplify the Expression

Now, simplify the right side of the equation: \(x = \frac{63}{-9}\). Calculate \(63 \div -9\), which results in \(-7\).

Step 4: State the Solution

After simplifying, we find that \(x = -7\). This is the value of \(x\) that satisfies the equation.

Key Concepts

Variable Isolation

In the process of solving linear equations, one of the most important steps is variable isolation. This means that we aim to have the variable, often represented as \(x\), on one side of the equation all by itself. This allows us to easily identify its value. In the given equation, \(-9x = 63\), our "unknown" is stuck with \(-9\). By performing the same operation on both sides of the equation—in this case, dividing by \(-9\)—we can move one step closer to having \(x\) by itself.

• Consider what operation is currently acting on the variable. In this instance, \(x\) is being multiplied by \(-9\).
• To counteract this, the inverse operation is used: division. Hence, both sides of the equation are divided by \(-9\).
• This results in \(x = \frac{63}{-9}\).

Remember, the core goal is to transform the equation so that \(x\) remains alone, isolated on one side of the equation. This concept is key to solving for unknowns in linear equations.

Simplifying Expressions

Once the variable is isolated, the next step is simplifying any expressions that appear in the equation. Expressions are simplified to make them easier to understand or solve, often involving arithmetic operations like addition, subtraction, multiplication, or division. In this problem, after isolating the variable, we have the expression \(x = \frac{63}{-9}\). Simplifying this involves dividing \(63\) by \(-9\).

• Perform the division: \(63 \div -9 = -7\).
• The expression \(x = \frac{63}{-9}\) simplifies neatly to \(x = -7\).

Simplification can help make equations more manageable and solutions more apparent. This step ensures that the equation expresses \(x\) in its most straightforward form, leaving no ambiguity.

Negative Numbers

Dealing with negative numbers is crucial when working through linear equations, as they can significantly alter the sign and value of solutions. In this exercise, the equation starts with \(-9x = 63\), where \(-9\) is a negative number attached to the variable \(x\). This multiplicative factor needs particular attention when isolating the variable.

• When dividing the entire equation by \(-9\), keep in mind that a negative divided by a negative yields a positive result.
• Similarly, a positive divided by a negative gives a negative: hence \(\frac{63}{-9} = -7\).
• This step promptly uses our basic rule about signs, ensuring the solution has the correct sign.

Understanding how to manipulate negative numbers is fundamental, as it ensures accuracy in determining the sign—positive or negative—of your result. Mastery of this concept is necessary to solve equations correctly.