The sum of two numbers is 68, and one of them is 6 more than the other. What are the two numbers?
A. 29, 35
B. 31, 37
C. 32, 38
D. 30, 36
Explanation:
Let’s assume that one of the numbers is “x”. According to the problem statement, the other number is “6 more than the other”, or “x + 6”. The sum of these two numbers is 68. Therefore, we can set up an equation:
x + (x + 6) = 68
Simplifying this equation, we get:
2x + 6 = 68
Subtracting 6 from both sides, we get:
2x = 62
Dividing both sides by 2, we get:
x = 31
Now, we know that one of the numbers is 31. The other number is 6 more than this, or 37. Therefore, the two numbers are 31 and 37.
Therefore, the correct answer is B (31, 37).
Solving Word Problems Involving Linear Equations: Steps and Examples
Linear equations are used to solve a variety of word problems in mathematics. These problems can be solved by translating the given information into equations and then solving for the unknown quantity. Here are the steps to solve word problems involving linear equations:
Step 1: Read the problem carefully and identify the unknown quantity.
Step 2: Define variables for the unknown quantity and any other quantities mentioned in the problem.
Step 3: Translate the given information into equations.
Step 4: Solve the equations to find the value of the unknown quantity.
Step 5: Check your answer by substituting the value of the unknown quantity back into the original problem.
Let’s take an example to understand this better. Consider the problem:
“The sum of two numbers is 68, and one of them is 6 more than the other. What are the two numbers?”.
Step 1: The unknown quantity is the two numbers.
Step 2: Let x be one of the numbers. The other number is x + 6.
Step 3: The sum of these two numbers is 68, so we can write the equation:
x + (x + 6) = 68
Step 4: Simplifying this equation, we get:
2x + 6 = 68
Subtracting 6 from both sides, we get:
2x = 62
Dividing both sides by 2, we get:
x = 31
Therefore, one of the numbers is 31 and the other is x + 6, or 37.
Step 5: To check our answer, we can verify that the sum of these two numbers is indeed 68.
31 + 37 = 68
Therefore, our solution is correct.
In conclusion, solving word problems involving linear equations requires careful reading and understanding of the problem, defining variables, and translating the given information into equations. Once the equations are set up, we can solve for the unknown quantity using algebraic methods.
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