A Number Is 5 More Than 3 Times Another Number. The Sum Of The Two Numbers Is 33. As An Equation, This problem involves forming and solving algebraic equations to find the two numbers based on the given conditions. This type of problem is common in algebra and helps develop critical thinking and problem-solving skills. In this article, we'll explore how to translate word problems into algebraic equations, solve them step-by-step, and understand the practical applications of such problems.
Understanding the Word Problem
Before jumping into equations, it's essential to understand the problem statement clearly:
- First condition: One number (let's call it x) is 5 more than 3 times another number (let's call it y). Mathematically, this can be written as:
\[
x = 3y + 5
\]
- Second condition: The sum of the two numbers is 33:
\[
x + y = 33
\]
Our goal is to find the values of x and y that satisfy both conditions.
Forming the Equations
Based on the problem description, we can formulate two equations:
Equation 1: Relationship between x and y
\[
x = 3y + 5
\]
This equation states that x is 5 more than three times y.
Equation 2: Sum of the two numbers
\[
x + y = 33
\]
This indicates that the total of x and y is 33.
Solving the System of Equations
Having formulated the equations, the next step is to solve for the unknowns. There are various methods for solving systems of equations, such as substitution, elimination, or graphing. For this problem, substitution is the most straightforward.
Step 1: Substitute the expression for x into the second equation
Since from Equation 1, x = 3y + 5, substitute this into the second equation:
\[
(3y + 5) + y = 33
\]
Step 2: Simplify and solve for y
Combine like terms:
\[
3y + y + 5 = 33
\]
\[
4y + 5 = 33
\]
Subtract 5 from both sides:
\[
4y = 33 - 5
\]
\[
4y = 28
\]
Divide both sides by 4:
\[
y = \frac{28}{4} = 7
\]
So, the second number (y) is 7.
Step 3: Find x
Use the value of y in Equation 1:
\[
x = 3(7) + 5 = 21 + 5 = 26
\]
Thus, the first number (x) is 26.
Verification of the Solution
It's important to verify whether the solution satisfies both original conditions.
- Check the first condition: Is x 5 more than 3 times y?
\[
3 \times 7 + 5 = 21 + 5 = 26
\]
Yes, x equals 26, which matches the calculated value.
- Check the second condition: Is the sum of x and y 33?
\[
26 + 7 = 33
\]
Yes, the sum is correct.
Therefore, the solution is:
\[
x = 26, \quad y = 7
\]
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Practical Applications of Such Algebraic Problems
Understanding how to formulate and solve equations based on word problems has numerous real-world applications. These include:
- Financial Planning: Calculating savings, investments, or loan payments based on given conditions.
- Business and Economics: Determining profit margins, costs, and revenues from textual data.
- Engineering: Solving for unknown parameters in design problems.
- Statistics: Interpreting data and forming models to predict outcomes.
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Additional Example: Variations of the Problem
To deepen understanding, consider how similar problems can be structured and solved.
Example 1: Different sum or relationships
Suppose the sum of the two numbers is 50, and one number is 4 more than twice the other.
- Define x and y as before.
- Conditions:
\[
x = 2y + 4
\]
\[
x + y = 50
\]
Solution:
Substitute x into the second equation:
\[
(2y + 4) + y = 50 \Rightarrow 3y + 4 = 50
\]
\[
3y = 46 \Rightarrow y = \frac{46}{3} \approx 15.33
\]
Then:
\[
x = 2 \times \frac{46}{3} + 4 = \frac{92}{3} + 4 = \frac{92}{3} + \frac{12}{3} = \frac{104}{3} \approx 34.67
\]
This example demonstrates that solutions can be fractional when the problem parameters are different.
Example 2: Introducing inequalities
Suppose you're asked to find numbers where one is 5 more than three times the other, and their sum exceeds 30.
- Conditions:
\[
x = 3y + 5
\]
\[
x + y > 30
\]
- Solving for specific values involves similar steps but considering inequalities.
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Tips for Solving Algebraic Word Problems
When approaching problems like this, keep in mind the following tips:
- Identify variables: Assign symbols to unknown quantities.
- Translate words into equations: Carefully interpret the language and convert it into algebraic expressions.
- Choose an appropriate method: Use substitution, elimination, or graphing based on the problem.
- Verify solutions: Always check whether your answers satisfy the original conditions.
- Practice with varied problems: Exposure to different scenarios improves problem-solving skills.
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Conclusion
Solving algebraic problems based on word descriptions, such as "A number is 5 more than 3 times another number, and their sum is 33," enhances critical thinking and mathematical reasoning. By translating words into equations, simplifying, and solving systematically, you can efficiently find the unknowns. These skills are vital not only in academics but also in everyday life, where problem-solving and analytical thinking are required. Continue practicing similar problems to strengthen your understanding of algebra and develop confidence in tackling various mathematical challenges.
