Problem 30 - Fundamentals of Mathematics

Question

Solve each equation. Be sure to check each result. $$ 5-11 x=27 $$

Step-by-Step Solution

Answer

The solution is \(x = -2\).

Step 1: Isolate the Variable

Begin by isolating the variable \(x\) on one side of the equation. Currently, you have the equation \(5 - 11x = 27\). To isolate \(x\), first subtract 5 from both sides:\[5 - 11x - 5 = 27 - 5\]Which simplifies to:\[-11x = 22\]

Step 2: Solve for x

Now that we have \(-11x = 22\), divide both sides of the equation by \(-11\) to solve for \(x\):\[x = \frac{22}{-11}\]Simplifying this gives:\[x = -2\]

Step 3: Check the Solution

To ensure that \(x = -2\) is correct, substitute it back into the original equation and verify the result matches:\[5 - 11(-2) = 27\]Simplifying the left side:\[5 + 22 = 27\]This simplifies to \(27 = 27\), confirming the solution is correct.

Key Concepts

Isolating the Variable

Whenever you're working with linear equations, one of the most critical tasks is isolating the variable. In our equation, \(5 - 11x = 27\), the goal is to get \(x\) by itself on one side of the equation. To do this effectively:

• Look at the terms on each side. Notice that \(-11x\) is attached to the 5 on the left side.
• Use inverse operations to untangle the variable. Here, you subtract 5 from both sides to get rid of it next to \(-11x\).
• After subtracting, you have \(-11x = 22\), making \(x\) more isolated and manageable.

The idea is to simplify step by step until \(x\) stands alone. This sets the stage for solving the equation accurately.

Checking Solutions

Once you think you have the right answer, you need to check that your solution is correct. This seems like an extra step at first, but it's crucial in ensuring you've solved the equation accurately.Here's how you check:

• After calculating \(x = -2\), plug \(x\) back into the original equation \(5 - 11x = 27\).
• Substituting gives \(5 - 11(-2)\).
• Simplify the expression: \(5 + 22 = 27\).

If both sides of the equation are equal, like \(27 = 27\), this confirms that \(x = -2\) is indeed the correct solution. Checking ensures you haven't made any mistakes.

Simplifying Equations

The art of solving linear equations is deeply connected with simplification. Simplifying helps clear the clutter so you can see what the equation really wants to tell you. Here's a glance at the steps:

• Firstly, simplify each side of the equation individually. This can be done by combining like terms or distributing values if needed.
• Next, focus on eliminating any addition or subtraction around the variable, as seen when we subtract 5 from both sides.
• Finally, divide or multiply as necessary to get the variable by itself. We divided by \(-11\) in this exercise.

Simplification gives clarity and precision, crucial for correctly solving any linear equation. By consistently simplifying, you keep equations easy to work through.