Problem 13 - Elementary Algebra Within Reach

Question

Solve the equation and check your solution. $$3-(2 x-18)=3$$

Step-by-Step Solution

Answer

The solution of the equation is \(x = 9\).

Step 1: Remove Brackets

Start by removing the brackets and simplify the equation. In the expression 3 - (2x-18), distribute the negative sign into the bracket which transforms the equation into 3 - 2x + 18 = 3.

Step 2: Simplify

Next, combine like terms on the same side of the equation. This results in 21 - 2x = 3.

Step 3: Isolate the variable

Rearrange the equation to isolate the variable x. Substract 21 from both sides to get -2x = 3 - 21 which simplifies to -2x = -18.

Step 4: Solve for x

Next, divide both sides of the equation by -2 to solve for x. Solve the equation -2x = -18 by dividing both sides by -2 to get x = -18/(-2) which simplifies to x = 9.

Step 5: Verify the answer

Lastly, substitute x = 9 into the original equation 3-(2 x-18) = 3 and check if both sides of the equation are equal. When you substitute x = 9 to original equation, it simplifies to 3 - (2(9) -18) = 3 which simplifies to 3 - 0 = 3 verifying that x = 9 is indeed the correct solution.

Key Concepts

Algebraic Expressions

Algebraic expressions are foundational in solving equations. When you look at an expression like \(3 - (2x-18)\), it involves variables, numbers, and operations grouped together. Understanding how to manipulate these expressions is crucial.

• **Variables:** These are symbols, often letters, used to represent unknown values. In our example, \(x\) is the variable.
• **Numbers:** These are constants in the expression, such as 3 and -18.
• **Operations:** These include addition, subtraction, multiplication, and division, as seen in the expression.

In solving equations, you'll frequently encounter expressions that require simplification. This often involves distributing negative signs across terms inside brackets, as done here, turning \(3 - (2x-18)\) into \(3 - 2x + 18\). By rephrasing parts of the expression, you prepare the equation for solving. Always remember to keep track of all changes to maintain the integrity of the original equation.

Isolating Variables

Isolating variables means rearranging the equation so that the variable stands alone on one side. This step is key to finding the value of the variable.

How to Isolate a Variable:

• **Combine Like Terms:** Simplify both sides of the equation as much as possible. For instance, we combined terms in \(21 - 2x = 3\).
• **Inverse Operations:** Use operations that "undo" each other to isolate the variable. This involves addition and subtraction, multiplication or division, depending on what will best "cancel out" the surrounding numbers.
• **Solving:** In the equation \(-2x = -18\), dividing both sides by \(-2\) isolates \(x\), giving \(x = 9\).

This procedure may involve several steps but always moves you toward the ultimate goal: finding the value of the unknown variable. Isolating variables involves balance; whatever you do to one side of the equation, you must do to the other, ensuring the equation remains equal.

Verifying Solutions

Once you determine a solution, in this case \(x = 9\), it’s important to verify it, ensuring no errors occurred during solving processes. Verification proves that your solution makes the original equation true.

Steps to Verify a Solution:

• **Substitute the Solution:** Plug the found value of \(x\) back into the original equation. Here, substituting gives \(3 - (2(9) - 18) = 3\).
• **Simplify:** Break down the equation. Check whether both sides equal each other. Simplifying gives us \(3 - 0 = 3\).
• **Check Equality:** If both sides equal each other correctly, your solution is verified. In this case, it's indeed \(3 = 3\).

Verification is crucial, as it boosts confidence in the correctness of your solution. Each step provides insight into whether you've manipulated the equation correctly and fully.