what multiplies to and adds to 7

Natashya

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27-12-2024

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This article will guide you through finding pairs of numbers that satisfy specific multiplication and addition conditions. We'll explore the process of finding two numbers that multiply to a target product (in this case, 7) and add to a target sum (in this case, 8). This is a common problem in algebra and number theory.

Understanding the Problem

The core question is: What two numbers, when multiplied together, equal 7, and when added together, equal 8?

Let's break down the problem:

• Multiplication: We need two numbers whose product is 7.
• Addition: These same two numbers must add up to 8.

Finding the Solution: A Systematic Approach

We can solve this problem using a few different approaches:

1. Listing Factors:

Since 7 is a prime number (only divisible by 1 and itself), its only integer factors are 1 and 7. Let's check if these satisfy the addition condition:

• 1 + 7 = 8

This works! Therefore, the two numbers are 1 and 7.

2. Algebraic Approach:

We can also solve this using algebra. Let's represent the two unknown numbers as 'x' and 'y'. We can set up a system of two equations:

• Equation 1 (Multiplication): x * y = 7
• Equation 2 (Addition): x + y = 8

Solving this system of equations requires some algebraic manipulation. One way to do this is to solve Equation 2 for one variable (e.g., solve for y: y = 8 - x) and substitute this expression into Equation 1:

x * (8 - x) = 7

Expanding and rearranging this equation gives us a quadratic equation:

8x - x² = 7

x² - 8x + 7 = 0

This quadratic equation can be factored as:

(x - 1)(x - 7) = 0

This gives us two possible solutions for x: x = 1 and x = 7.

If x = 1, then y = 8 - 1 = 7. If x = 7, then y = 8 - 7 = 1.

Both solutions lead to the same pair of numbers: 1 and 7.

Extension: What if the numbers were different?

Let's consider a slightly different problem: What two numbers multiply to 12 and add to 7?

This problem is more complex because 12 has multiple factor pairs. Let's list them:

• 1 and 12
• 2 and 6
• 3 and 4

Now let's check which pair adds up to 7:

• 1 + 12 = 13
• 2 + 6 = 8
• 3 + 4 = 7

The pair that works is 3 and 4.

Conclusion

Finding pairs of numbers that satisfy both multiplication and addition conditions involves a combination of understanding factors and using algebraic techniques when necessary. The method you choose will depend on the complexity of the problem. Remember, prime numbers simplify the process significantly! The original question, "What two numbers multiply to 7 and add to 8?", has a solution of 1 and 7.