Problem 26 - Elementary Algebra

Question

Solve. $$ -x=-10 $$

Step-by-Step Solution

Answer

x = 10

Step 1: Understand the Equation

The given equation is \(-x = -10\). The expression \(-x\) means that \(x\) is being multiplied by \(-1\). Therefore, our goal is to solve for \(x\) by isolating it.

Step 2: Isolate x

To isolate \(x\), we need to remove the negative sign. We do this by multiplying both sides of the equation by \(-1\). This will cancel out the negative sign on \(-x\).

Step 3: Simplify Both Sides

Multiply both sides by \(-1\): \[-1(-x) = -1(-10)\]This simplifies to:\[x = 10\]

Key Concepts

Negative Numbers

Understanding negative numbers is essential in solving linear equations like the one given in the exercise. A negative number is any number less than zero, usually found to the left of zero on the number line. It is represented by a minus sign (−) before the number. In our equation, we see

• directly translating the equation \(-x = -10\) which implies that the term \(x\) is being negated or multiplied by -1. In simpler terms, if flipping signs, if applied to both counterparts of the equation, you often switch numbers from negative to positive (or vice versa) to maintain the balance the equation represents.

Recognizing these characteristics helps you to effectively manage and balance equations that feature negative numbers.

Multiplication

When dealing with the multiplication of numbers, whether positive or negative, it is crucial to apply the multiplication rules correctly. Multiplication by -1 is a common step in solving equations with variables that have a negative sign in front of them. It helps to remove the negative sign and works according to a simple rule:

• A negative times a negative results in a positive number.
• A positive times a negative results in a negative number.

In the equation given, \(-x = -10\), multiplying both sides by -1 transforms it: \[-1(-x) = -1(-10)\]. This way, we cancel the negative sign from the left side, thus getting a clean positive version of x y.

Simplification

Simplification is the process of making an equation or expression easier to understand or use. It often involves combining like terms, reducing fractions, or eliminating unnecessary elements. It allows the equation to become clearer and keeps it in its simplest form for solving. In our context, simplifying involves making the expression \(-x = -10\) more straightforward to solve. By executing our previous multiplication step: \[-1(-x) = -1(-10)\], we simplify the equation:

• We change \(-x\) to x
• We change \(-10\) to 10

So, it reads simply as x = 10, which is much easier to interpret and solve.

Isolation of Variables

The isolation of variables is the process of manipulating an equation so that the variable (often denoted by x) stands alone on one side of the equation. The primary goal is to clearly solve for x. This is accomplished by performing operations that cancel out other terms attached to x.In our exercise, the variable is encumbered by a negative sign, represented as \(-x\).To isolate x, we applied the multiplication by -1 to both sides of the equation, effectively reducing the equation to:

• \(x = 10\)

This operation allowed x to be on its own, enabling us to see the solution more clearly and quickly. This makes isolation of variables a fundamental step when solving such equations.