Problem 34 - Elementary Algebra

Question

Solve. $$ -t 12=14 $$

Step-by-Step Solution

Answer

The value of \(t\) is \(-14\).

Step 1: Identify the equation and the unknown

The equation given is \(-t = 12 = 14\). This appears to be a typographical error in the equation, as it simultaneously states \(-t = 12\) and equals 14, which is not possible. Thus, we will solve this based on the typical understanding, which is \(-t = 14\). Our task is to find the value of \(t\).

Step 2: Isolate the variable

Since the equation is \(-t = 14\), we will multiply both sides by \(-1\) to isolate \(t\). This simplifies the equation to \(t = -14\).

Step 3: Verify the solution

Check if substituting \(-14\) back into the original equation \(-t = 14\) holds true. Substitute \(t = -14\), then \(-(-14) = 14\), which simplifies to \(14 = 14\). Thus, the solution is correct.

Key Concepts

Equation Solving

Solving an equation is like unraveling a puzzle. You have an equation that asserts two expressions are equal using an equals sign (=). The objective is to find the value of a variable that makes this true.

Let's consider the example given: the equation is actually meant to be \(-t = 14\). To solve this, you need to find the value of \(t\) that satisfies the equation. The steps are:

• Identify and understand the equation and what it represents.
• Perform operations to both sides to maintain equality.

Equation solving requires an understanding of operations and how to reverse them. Multiplication, division, addition, and subtraction are your tools to "undo" operations applied to the variable.

Negative Numbers

Negative numbers might seem puzzling at first, but they're simply values less than zero. The sign in front of a number indicates its direction on the number line.

In the equation \(-t = 14\), the minus sign in front of \(t\) means that \(t\) itself is negative. Here’s how you handle such a situation:

• A negative sign in front of a variable indicates the opposite of that variable.
• Multiplying or dividing by \(-1\) helps convert the negative variable into its positive counterpart, or vice versa.

By understanding negative numbers, we manage to keep our operations accurate. They tell us about the "direction" of these values on a number scale.

Variable Isolation

The goal of variable isolation is to "free" the variable from other terms to find its value. In the equation \(-t = 14\), isolating \(t\) involves changing its sign.

To isolate a variable:

• Identify the variable you need to isolate.
• Use inverse operations to cancel out other numbers or negative signs around it.
• Apply the same operation to both sides to maintain balance.

In this example, multiplying both sides by \(-1\) neutralizes the negative and isolates \(t\) as \(t = -14\).

Variable isolation is a fundamental skill in algebra, helping you solve equations and understand relationships between numbers.