For students who have completed Lesson 15 Homework 5-1, it is time to check your work and find out if you have the correct answers. In this answer key, we will provide a detailed explanation for each question and offer a step-by-step guide to help you understand the concept behind each solution.
Lesson 15 Homework 5-1 focuses on a variety of topics, including algebraic expressions, equations, and inequalities. By solving these problems, students can practice their skills in simplifying expressions, solving equations with fractions, and graphing linear inequalities. Through a thorough understanding of these concepts, students will be better equipped to tackle more advanced math problems in the future.
In this answer key, we will go over each question and provide the correct answer along with an explanation of how to arrive at that answer. We understand that math can sometimes be challenging, but with a clear understanding of the concepts and a step-by-step approach, any student can successfully complete Lesson 15 Homework 5-1.
Lesson 15 Homework 5.1 Answer Key
In this lesson’s homework, we will review the answer key for Exercise 5.1. This exercise focuses on solving algebraic equations involving multiplication and division.
Question 1:
Mr. Smith has 8 boxes of apples. Each box contains 6 apples. How many apples does Mr. Smith have in total?
To solve this question, we need to multiply the number of boxes by the number of apples per box. In this case, we multiply 8 by 6:
- 8 x 6 = 48
Therefore, Mr. Smith has a total of 48 apples.
Question 2:
Sara wants to share 20 candies equally between herself and 4 friends. How many candies will each person receive?
To solve this question, we need to divide the total number of candies by the number of people. In this case, we divide 20 by 5 (Sara and her 4 friends):
- 20 ÷ 5 = 4
Therefore, each person will receive 4 candies.
These are just two examples of the types of questions covered in Exercise 5.1. By practicing this exercise, students can strengthen their skills in solving multiplication and division equations.
Understanding the Problem
When faced with a problem, it is essential to first understand its nature and scope. This involves carefully analyzing the problem statement, identifying key elements, and determining the desired outcome. By gaining a clear understanding of the problem, we can effectively develop a solution that addresses its root cause and meets the needs of the stakeholders.
One approach to understanding the problem is to break it down into smaller components or sub-problems. This can help to identify any underlying issues and dependencies, as well as determine the sequence of steps required to solve the problem as a whole. By breaking the problem down into manageable parts, we can approach each sub-problem individually and systematically work towards a solution.
Another important aspect of understanding the problem is to gather and analyze relevant data. This could involve conducting research, collecting information from stakeholders, or studying similar problems and their solutions. By gathering data, we can gain insights into the problem’s context, root causes, and potential solutions. This information can then be used to inform the development of an effective solution.
In addition to analyzing the problem and gathering data, it is crucial to consider the perspective of the stakeholders. Understanding their needs, expectations, and constraints can help shape the solution and ensure its alignment with their interests. This may involve conducting interviews, surveys, or workshops to gather input and involve stakeholders in the problem-solving process.
Overall, understanding the problem is a critical step in problem-solving. It allows us to gain clarity on the problem’s nature, break it down into manageable parts, gather and analyze relevant data, and consider the needs of the stakeholders. By taking the time to thoroughly understand the problem, we can develop more effective and tailored solutions that address its underlying causes and meet the needs of those involved.
Steps to Solve Homework 5.1
Homework 5.1 is a set of exercises that require careful analysis and problem-solving skills. Here are the steps you can follow to solve the homework effectively:
Step 1: Read the Problem Statement
The first step is to carefully read and understand the problem statement. Pay attention to the given information, any constraints or requirements, and the objective of the problem.
Step 2: Identify the Known and Unknown Variables
After understanding the problem, identify the known and unknown variables. Known variables are the values or quantities that are given in the problem statement, while unknown variables are the values or quantities you need to find.
Step 3: Plan a Solution Strategy
Based on the known and unknown variables, plan a solution strategy. This could involve using formulas, equations, or logical reasoning to determine the relationships between the variables and develop a step-by-step approach to solve the problem.
Step 4: Solve the Problem
Implement your solution strategy to solve the problem. Perform the necessary calculations, substitutions, or logical deductions to find the values of the unknown variables. Keep track of your steps and double-check your work to ensure accuracy.
Step 5: Verify the Solution
Once you have found the values of the unknown variables, verify your solution. Check if the calculated values satisfy the given constraints or requirements in the problem statement. Ensure that your solution makes sense logically and mathematically.
Step 6: Document Your Solution
Finally, document your solution clearly and concisely. Write down the calculations, steps, and final answers in an organized manner. Use proper notation and labels to make your solution easy to understand and follow.
By following these steps, you can effectively solve Homework 5.1 and demonstrate your understanding of the concepts and problem-solving skills.
Solution for Question 1
To solve this inequality, we can start by isolating the variable (x) on one side of the inequality sign. We do this by adding 5 to both sides of the equation:
(7x – 5 + 5 leq 12 + 5)
This simplifies to:
(7x leq 17)
Next, we divide both sides of the inequality by 7 to solve for (x). Since we are dividing by a positive number, the inequality sign does not change:
(frac{7x}{7} leq frac{17}{7})
This simplifies to:
(x leq frac{17}{7})
The solution for the inequality (7x – 5 leq 12) is (x leq frac{17}{7}).
Solution for Question 2
The second question asks for the equations of the two lines that form the diagonals of a parallelogram. To find these equations, we first need to determine the slopes of the two lines.
We know that the opposite sides of a parallelogram are parallel, which means that the slopes of the two lines are equal. So, we need to find the slope of one of the diagonals in order to determine the slope of the other diagonal.
Let’s take one pair of opposite vertices of the parallelogram, A and C, and find the slope of the line passing through them. We can use the formula:
slope = (y2 – y1) / (x2 – x1)
If we plug in the coordinates of points A(3, -4) and C(7, 2), we can calculate the slope of the line passing through them:
slope = (2 – (-4)) / (7 – 3) = 6 / 4 = 3/2
The slope of the line passing through A and C is 3/2. Since the diagonals of a parallelogram intersect at their midpoints, we can use the midpoint formula to find the coordinates of the midpoint of the other diagonal.
Once we have the slope and the midpoint, we can use the point-slope form of a line to find the equation of the second diagonal. The point-slope form is:
y – y1 = m(x – x1)
Using the midpoint as (x1, y1) and the slope as m, we can substitute these values into the equation and simplify to find the equation of the second diagonal.
Solution for Question 3
Question 3 asks about the factors that contributed to the outbreak of the First World War. Several key factors can be identified:
- Alliances: One of the main factors was the system of alliances between countries. These alliances created a complex web of commitments and obligations, which meant that when one country was attacked, its allies were automatically drawn into the conflict. For example, the assassination of Archduke Franz Ferdinand of Austria-Hungary by a Serbian nationalist in 1914 triggered the activation of various alliances, ultimately leading to the outbreak of war.
- Militarism: Another significant factor was the widespread militarism in Europe at the time. Many countries were engaged in an arms race, continuously building up their militaries and developing new weapons. This arms race created an atmosphere of tension and competition, heightening the risk of conflict.
- Imperialism: The quest for colonies and resources also played a role in the outbreak of war. European powers were engaged in intense competition for territories and resources around the world. The desire to expand and assert dominance led to conflicts and rivalries between nations.
- Nationalism: Nationalism, the belief in the superiority of one’s own nation and the desire to protect and promote its interests, also contributed to the outbreak of war. Nationalist sentiments were high in many countries, and the desire for territorial expansion and political autonomy fueled tensions between nations.
- Failure of diplomacy: Finally, the failure of diplomatic efforts to resolve conflicts peacefully played a significant role. Despite efforts to negotiate and mediate disputes, the countries involved were unable to reach satisfactory agreements, and diplomatic tensions escalated into war.
Overall, a complex interplay of alliances, militarism, imperialism, nationalism, and diplomatic failures contributed to the outbreak of the First World War. These factors created an environment of tension and competition, ultimately leading to the eruption of a devastating global conflict.
Solution for Question 4
Question 4 of the homework requires us to find the solution for a given equation. Let’s take a look at the equation:
2x + 3 = 7
To solve this equation, we need to isolate the variable x. Here are the steps:
- Start by subtracting 3 from both sides of the equation:
- 2x + 3 – 3 = 7 – 3
- 2x = 4
- Next, divide both sides of the equation by 2 to solve for x:
- 2x / 2 = 4 / 2
- x = 2
Therefore, the solution to the equation 2x + 3 = 7 is x = 2.