Problem 44 - Algebra 1: Concepts and Skills

Question

Write the sentence as an equation. Let x represent the number. Use mental math to solve the equation. Then check your solution. (Lesson 1.5).The sum of a number and 18 is 45.

Step-by-Step Solution

Answer

The unknown number (x) is 27.

Step 1: Translate the sentence into an equation

The sentence'related to the exercise can be written as an equation. 'The sum of a number and 18 is 45' can be expressed as: x + 18 = 45, where x represents the unknown number.

Step 2: Solve for x

In order to find the value of x, we should isolate x. This can be done by subtracting 18 from both sides of the equation: x + 18 - 18 = 45 - 18. Simplifying this equation yields x = 27

Step 3: Verify your solution

The last step is to check if our solution (x = 27) is correct. We substitute x = 27 into the original equation and check if it holds true: 27 + 18 = 45. Since the left side of the equation equals the right side, it confirms that the solution is correct.

Key Concepts

Solving Simple Algebraic Equations

Understanding how to solve simple algebraic equations is an essential skill in mathematics. A common scenario might involve a situation like the textbook exercise provided here: finding an unknown number, represented by a variable, which when added to 18 yields 45. The translation of this word problem into a mathematical expression involves setting up an equation like so:
\( x + 18 = 45 \).

To solve for the unknown \( x \), we use basic algebraic techniques – subtracting 18 from both sides to isolate the variable. The simplicity of the equation affords us the use of mental math. The solution \( x = 27 \) reveals the number that fulfills the condition in the original statement. When solving these types of equations, the goal is to perform operations that will 'undo' what has been done to the variable, allowing us to find its value. It's crucial to work through the order of operations in reverse; here, subtraction undoes addition. Remaining mindful of these methods not only clarifies the process but also equips you with the abilities to tackle more complex equations confidently.

Mental Math Strategies

Mental math can often feel like a magic trick, but it's really about practice and employing certain strategies to simplify calculations. When confronted with an equation such as \( x + 18 = 45 \), instead of reaching for a calculator, consider breaking down the problem. You recognize that 18 is close to 20, a friendly number to calculate with.

Approximation

Conceptually, if you add 20 to a number and get 45, that number would be 25. Since you 'overestimated' by 2, you adjust by subtracting 2 from 25 to get 23—the close neighbor to our actual answer of 27. This habit of rounding to the nearest 'round' number is a handy mental math strategy.

Subtraction in Parts

Another strategy is breaking the subtraction into parts. You can subtract 10 from 45 to quickly get to 35, then subtract the remaining 8 to land on 27. These techniques, among others, help improve your numerical fluency and can be particularly useful in everyday life where you need to make quick calculations.

Verifying Solutions in Algebra

Once you've calculated a potential solution to an algebraic equation, it’s important not to assume the work is done. The verification step is key to ensuring that your solution is correct. Using the example of our exercise, where we found that \( x = 27 \), how do we confirm this is correct? Insert your solution back into the original equation:

\( 27 + 18 = 45 \).

Upon evaluating the left side, we notice it does indeed equal 45, which matches the right side of the equation. If we had obtained a different number, it would indicate an error in our calculations.

Why Verification Matters

Verifying reinforces not only the accuracy of your answer but also your understanding of the relationship between the components of the equation. It is a vital step in solving algebraic problems and constitutes good mathematical practice. Always set aside time to check your solutions – it's the hallmark of thorough problem-solving and high-quality math work.