Problem 68 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$-3 x=21$$

Step-by-Step Solution

Answer

The solution to the equation \(-3x = 21\) is \(x = -7\).

Step 1: Identify the Equation

The equation presented in the problem is \(-3x = 21\). The aim is to isolate 'x' on one side of the equation.

Step 2: Apply Inverse Operations

To separate 'x' from '-3', multiply the both sides of the equation by \(-1/3\). Since \(-3 * -1/3\) equals to 1, it leaves 'x' alone on the left side of the equation. Similarly, \(21 * -1/3 = -7\) on the right hand side of the equation. Hence, the equation becomes \(x = -7\).

Step 3: Check Your Solution

To verify the solution, substitute 'x' with \(-7\) in the original equation \(-3 * -7 = 21\). Since the result is true, the solution is verified

Key Concepts

Inverse Operations

When solving linear equations, inverse operations are your best friend. They help you "undo" what's being done to the variable. This is crucial because it gets you one step closer to finding the solution. Let's break it down.

In our exercise, the equation was \(-3x = 21\). The variable \(x\) is being multiplied by \(-3\). To solve for \(x\), we need to do the opposite of multiplication, which is division. We divide both sides by \(-3\), effectively applying the inverse operation.

This is because:

• Multiplying and dividing are inverse operations.
• They cancel each other out, leaving you with the variable alone.

By calculating \(-3 \times -1/3\), we simplify it to 1, thus isolating \(x\) by itself. This concept of inverse operations ensures the equation maintains its balance without altering its equality.

Isolation of Variables

Isolating the variable is the main goal when you're solving an equation. You want the variable, \(x\), alone on one side of the equation.

Imagine you have a balance scale. Your job is to adjust the weights so only \(x\) remains on one side. In our equation \(-3x = 21\), after using inverse operations, we are left with \(x = -7\). Here's why isolation is key:

• You simplify the equation to find the exact value of the variable.
• It helps in understanding how the variable interacts with the constants and coefficients.

Isolation makes the solution crystal clear, like unraveling a mystery. Once you get the hang of it, your confidence in solving math problems will skyrocket!

Verification of Solutions

Verification ensures that the solution obtained is correct. It's a crucial step that cannot be skipped. To verify, you substitute the obtained value back into the original equation.

In our example, we found \(x = -7\). We plug this back into the original equation \(-3 \times -7 = 21\). Doing the multiplication gives us 21, which matches the right-hand side of the equation. This means our solution is spot on.

Verification confirms:

• The accuracy of your solution.
• That all mathematical steps have been correctly followed.
• It serves as a confidence check for your work!

Always remember, verifying solutions isn't just about being meticulous—it's about ensuring you've understood and solved the problem correctly. Keep practicing, and you'll find this process becoming second nature.