Problem 5 - Elementary Algebra

Question

Solve and check each equation. $$ 16 x-3-15 x=8 \text { for } x. $$

Step-by-Step Solution

Answer

Answer: x = 11

Step 1: Simplify both sides of the equation if possible

Combine like terms, if any. In this case, we simplify both sides: $$ 16x - 3 - 15x = 8 $$

Step 2: Isolate the variable x

Combine the terms of x on the left side and move the constant term to the right side: $$ 16x - 15x = 8 + 3 $$ $$ x = 11 $$

Step 3: Check the answer

Substitute the value of x we found (x = 11) back into the original equation to verify if the equation holds true: $$ 16(11) - 3 - 15(11) = 8 $$ $$ 176 - 3 - 165 = 8 $$ $$ 173 - 165 = 8 $$ $$ 8=8 $$ The left side equals the right side, so our solution is correct. The value of x that makes the equation hold true is x = 11.

Key Concepts

Solving Equations

Solving equations is like figuring out a mystery step by step. When you face an equation, you are trying to find out what number a variable—often represented by \( x \)—should be to make that equation true. Let's break it down simply.

The first step is to simplify the equation if possible. This means looking at both sides and combining any like terms. Like terms are terms that have the same variables raised to the same power. For instance, in our exercise:

• The terms \( 16x \) and \( -15x \) are like terms because they both contain \( x \).

By combining them, you make the equation simpler. Here, \( 16x - 15x \) turns into \( x \). Now the equation is easier to work with, making the next steps much clearer.

Like Terms

Like terms are your friends when making equations simpler. They allow you to combine similar elements in the equation. This is crucial because it reduces the complexity and helps you to find the variable's value faster. In mathematical terms:

• Like terms are terms that have the same variable part, like \( 2x \) and \( 3x \).
• They can also be numbers, such as combining \( -3 \) and \( +3 \) as integers.

In our exercise, combining \( 16x \) and \( -15x \) gives \( x \), which is the process of simplifying by using like terms. After simplifying, the equation becomes much easier to handle, paving the way to find the solution.

Check Solutions

After solving an equation, it is important to check your work. This ensures that the answer you found truly satisfies the original equation. Here's how you can do it:

Take the solution you've found and substitute it back into the original equation. For our exercise, if \( x = 11 \), substitute 11 everywhere you see \( x \) in the original equation:

• \( 16(11) - 3 - 15(11) =? 8 \)

Calculate step by step:

• \( 176 - 3 - 165 = 8 \)
• \( 173 - 165 = 8 \)
• Finally, \( 8 = 8 \), confirming that our solution is correct.

This shows that the value we calculated for \( x \) works perfectly, ensuring we've solved the equation correctly. Checking solutions helps you confirm that no small errors occurred in your calculation steps.