Problem 88 - Algebra A Combined Function

Question

Fill in the chart by finding two numbers that have the given product and sum. The first column is filled in for you. $$ \begin{array}{|l|c|c|c|c|c|c|c|c|} \hline & & \text { 85. } & \text { 86. } & \text { 87. } & \text { 88. } & \text { 89. } & \text { 90. } & \text { 91. } & \text { 92. } \\ \hline \text { Two Numbers } & 4,7 & & & & & & & & \\ \hline \text { Their Product } & 28 & 12 & 20 & 8 & 16 & -10 & -9 & -24 & -36 \\\ \hline \text { Their Sum } & 11 & 8 & 9 & -9 & -10 & 3 & 0 & -5 & -5 \\ \hline \end{array} $$

Step-by-Step Solution

Answer

The pairs are (2,6), (4,5), (-1,-8), (-2,-8), (-2,5), (3,-3), (3,-8).

Step 1: Understand the problem

We need to find two numbers for each exercise number (86 to 92) such that their product and sum match the given values in the chart.

Step 2: Solve Exercise 86

We are given a product of 12 and a sum of 8. Let's find two numbers that satisfy both conditions:- Start with two numbers, say \( x \) and \( y \), then \( x \times y = 12 \) and \( x + y = 8 \).- Factorize 12 into pairs: (1, 12), (2, 6), (3, 4), etc.- The pair (2, 6) satisfies \( 2 \times 6 = 12 \) and \( 2 + 6 = 8 \).- So, the two numbers are 2 and 6.

Step 3: Solve Exercise 87

We are given a product of 20 and a sum of 9.- The equation setup is \( x \times y = 20 \) and \( x + y = 9 \).- Factorize 20 into pairs: (1, 20), (2, 10), (4, 5), etc.- The pair (4, 5) satisfies \( 4 \times 5 = 20 \) and \( 4 + 5 = 9 \).- The two numbers are 4 and 5.

Step 4: Solve Exercise 88

We have a product of 8 and a sum of -9.- Set it up as \( x \times y = 8 \) and \( x + y = -9 \).- Factorize 8 into pairs: (1, 8), (2, 4), etc.- With negative sums, consider negative factors: (-1, -8), (-2, -4), etc.- The pair (-1, -8) satisfies \( -1 \times -8 = 8 \) and \( -1 + (-8) = -9 \).- So, the numbers are -1 and -8.

Step 5: Solve Exercise 89

We have a product of 16 and a sum of -10.- The equation is \( x \times y = 16 \) and \( x + y = -10 \).- Possible factor pairs of 16 include: (1, 16), (2, 8), (4, 4), etc.- Include negative pairs: (-2, -8) satisfies \( -2 \times -8 = 16 \) and \( -2 + (-8) = -10 \).- Thus, the numbers are -2 and -8.

Step 6: Solve Exercise 90

We need a product of -10 and a sum of 3.- The setup is \( x \times y = -10 \) and \( x + y = 3 \).- Consider negative product pairs: (-1, 10), (1, -10), (-2, 5), (2, -5), etc.- The pair (-2, 5) matches as \( -2 \times 5 = -10 \) and \( -2 + 5 = 3 \).- Hence, the numbers are -2 and 5.

Step 7: Solve Exercise 91

We need a product of -9 and a sum of 0.- With \( x \times y = -9 \) and \( x + y = 0 \), the numbers must be opposites.- Consider pairs: (3, -3) fits since \( 3 \times -3 = -9 \) and \( 3 + (-3) = 0 \).- Therefore, the numbers are 3 and -3.

Step 8: Solve Exercise 92

We are looking for a product of -24 and a sum of -5.- We set \( x \times y = -24 \) and \( x + y = -5 \).- Examine factor pairs: (1, -24), (-1, 24), (3, -8), (8, -3), etc.- The pair (3, -8) fits as \( 3 \times -8 = -24 \) and \( 3 + (-8) = -5 \).- Thus, the numbers are 3 and -8.

Key Concepts

Factorization

Factorization is a powerful tool in algebra that involves breaking down a number into a product of other numbers, or "factors," that multiply together to produce the original number. This process is particularly useful when solving problems involving multiplication and addition, such as finding two numbers that meet specified product and sum conditions.

For example, if you're given a product of 12, you can factorize it into pairs such as (1, 12), (2, 6), and (3, 4). These pairs multiply to give the original product 12. Factorizing becomes even more interesting when we consider both positive and negative numbers, especially when dealing with negative products. In such cases, we consider pairs like (-1, -12), (-2, -6), and (-3, -4), which also multiply to give positive results when multiplied by each other.

Effective factorization requires practice and understanding of numbers. It's the foundation upon which solving simultaneous equations with products and sums becomes much easier. By systematically listing factor pairs, you can efficiently determine which pair fits both the product and sum criteria given in a problem.

Solving Equations

Solving equations in algebra often involves finding unknown values that satisfy certain conditions, often expressed in terms of sums and products. In the exercises given, we need to find two numbers, let's call them \( x \) and \( y \), that satisfy both a product \( x \times y \) and a sum \( x + y \).

When solving these problems:

• First, write down the equation for the product; for example, \( x \times y = 20 \).
• Next, write down the equation for the sum; for instance, \( x + y = 9 \).
• Identify factor pairs that satisfy the product condition.
• Check these factor pairs to see which one also meets the sum condition.

For instance, while solving, you have a product of 20, possible factor pairs might be (1, 20), (2, 10), and (4, 5). Upon checking which of these pairs adds up to 9, we find that (4, 5) is a perfect match. Thus, these figures are the solution.

By breaking down equations into simple steps and evaluating pairs, even complex-seeming algebraic equations can become accessible and straightforward.

Number Pairs

Number pairs are simply sets of two numbers that together satisfy specific mathematical conditions. In algebra, these conditions typically relate to a given product and sum that the number pairs must achieve.

In the given exercises, number pairs are essential in solving the question, as each problem asks for two numbers based on their product and sum values.

• For a product of 16 and a sum of -10, number pairs like (-2, -8) fulfill the criteria because they multiply to 16 and add up to -10.
• This knowledge helps identify valid pairs quickly.

Considerations for finding number pairs include examining both positive and negative values, especially when a sum or product could involve negative numbers or results that challenge straightforward logic.

Practicing with number pairs allows you to see connections between different operations such as addition and multiplication. This skill is crucial for strengthening overall mathematical problem-solving capabilities, as it encourages logical thinking and methodical processing of information.