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Chapter 11: Problem 13

Solve using the multiplication principle. Don't forget to check! $$ -12 x=72 $$

Short Answer

Expert verified
x = -6

Step by step solution

01

Isolate the variable

To solve the equation \(-12x = 72\), start by isolating the variable x. Divide both sides of the equation by -12.
02

Perform the division

Divide both sides by -12 to isolate x: \(-12x \div -12 = 72 \div -12\). This simplifies to \(x = -6\).
03

Check the solution

Substitute \(x = -6\) back into the original equation to verify: \(-12(-6) = 72\). Simplifying the left side: \72 = 72\, which is correct.

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Key Concepts

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

Solve Linear Equations
To solve linear equations like \(-12x = 72\), you need to find the value of the variable that makes the equation true. Linear equations usually have one solution that you can find by performing operations such as addition, subtraction, multiplication, and division. In this case, we use the multiplication principle to isolate the variable. Understanding the principles of solving linear equations helps in mathematics, science, and everyday problem-solving scenarios. Let's explore how to isolate the variable and check the solution step-by-step.
Isolate the Variable
Isolating the variable means getting the variable \(x\) on one side of the equation and everything else on the other side. This involves performing operations that 'undo' what鈥檚 been done to the variable. In our example, the variable \(x\) is multiplied by \(-12\). To isolate \(x\), we need to divide both sides by \(-12\):
    \[-12x = 72 \div -12 = -6\]
Notice that dividing both sides by \(-12\) cancels out the \(-12\) multiplied by \(x\) on the left side, leaving us with \(x = -6\). Thus, the variable is now isolated. This crucial step simplifies the equation and allows us to find the value of the variable.
Check the Solution
After solving for \(x\), it is important to check your solution to ensure it is correct. To do this, substitute the value back into the original equation and verify if both sides are equal. For our example:
    Substitute \(x = -6\): \[ -12(-6) = 72 \]
Simplify the left side:
    \[ 72 = 72 \]
Since both sides of the equation are equal, \(-12x = 72\) is verified as correctly solved, confirming that \(x = -6\) is indeed the correct solution. Checking your solutions helps catch errors and reinforces your understanding of the process.

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