Section 4 is a crucial part of Math Nation, as it tests your understanding and application of various mathematical concepts. In this article, we will provide the answers to the Test Yourself questions in Section 4, allowing you to check your work and evaluate your progress.
These Test Yourself questions cover a wide range of topics, including algebraic expressions, equations, inequalities, and functions. By completing these questions and reviewing the corresponding answers, you will solidify your understanding of these fundamental mathematical concepts.
Whether you are preparing for an upcoming exam, reviewing for a math competition, or simply seeking to improve your math skills, the answers to the Test Yourself questions in Section 4 will serve as a valuable resource. By comparing your answers to the correct ones, you can identify areas where you may need further practice and strengthen your overall mathematical proficiency.
Math Nation Section 4 Test Yourself Answers: Question 1
Substituting x = 4 into the equation, we get y = 3(4) – 9. Simplifying this expression, we have y = 12 – 9, which equals y = 3.
Therefore, when x is equal to 4, the value of y in the equation y = 3x – 9 is 3. This can be seen as the coordinates (4, 3) on the graph of the equation.
Explanation of Question 1 Answer
Question 1 asks: “What is the value of √(4^3 x 2^5)?”
To solve this question, we need to simplify the expression under the square root (√). We have 4^3 x 2^5. Let’s break it down:
- 4^3 means 4 raised to the power of 3. In other words, 4^3 = 4 x 4 x 4 = 64.
- 2^5 means 2 raised to the power of 5. In other words, 2^5 = 2 x 2 x 2 x 2 x 2 = 32.
Now, let’s substitute these values into the original expression:
√(4^3 x 2^5) = √(64 x 32)
Next, we can simplify the expression further by finding the product of 64 and 32:
√(64 x 32) = √2048
- √2048 can be simplified by finding the largest perfect square that is a factor of 2048. In this case, we know that 64 is a perfect square and a factor of 2048: 2048 ÷ 64 = 32.
We can rewrite √2048 as √(64 x 32) or √64 x √32. √64 is equal to 8, so we have:
√2048 = 8 x √32
Finally, we can simplify further by finding the largest perfect square that is a factor of 32. In this case, 16 is a perfect square and a factor of 32: 32 ÷ 16 = 2.
Substituting this value in, we get:
8 x √32 = 8 x 2√2 = 16√2
Therefore, the value of √(4^3 x 2^5) is 16√2.
Math Nation Section 4 Test Yourself Answers: Question 2
In Question 2 of the Math Nation Section 4 Test, we are given a quadratic equation in the standard form, which is:
2x^2 + 5x – 12 = 0
To solve this quadratic equation, we can use the factorization method or the quadratic formula. Let’s first try factoring the equation.
We need to find two numbers that multiply to give -24 (the product of the coefficient of x^2 and the constant term -12) and add up to give 5 (the coefficient of x). After some trial and error, we find that the numbers are 8 and -3, so we can rewrite the equation as:
2x^2 + 8x – 3x – 12 = 0
Now, we group the terms and factor by grouping:
2x(x + 4) – 3(x + 4) = 0
We can now factor out the common binomial (x + 4) to get:
(x + 4)(2x – 3) = 0
Now we can set each factor equal to zero and solve for x:
x + 4 = 0 or 2x – 3 = 0
Solving these equations, we find the two possible values of x:
- x = -4
- x = 3/2
Therefore, the solutions to the quadratic equation 2x^2 + 5x – 12 = 0 are x = -4 and x = 3/2.
Explanation of Question 2 Answer
The second question in the Math Nation Section 4 test asks about the area of a triangle. The question presents a triangle with a base of 6 units and a height of 8 units. The answer to this question can be found by using the formula for the area of a triangle: A = 1/2 * base * height. In this case, the base is 6 units and the height is 8 units, so the calculation would be: A = 1/2 * 6 * 8 = 24 units squared.
It is important to note that the formula for the area of a triangle is derived from the fact that a triangle can be seen as half of a rectangle. In this case, the base of the triangle is like the base of a rectangle, and the height of the triangle is like the height of a rectangle. By multiplying the base by the height and dividing by 2, we can find the area of the triangle.
Therefore, the answer to question 2 is 24 units squared, which represents the area of the given triangle with a base of 6 units and a height of 8 units.
Math Nation Section 4 Test Yourself Answers: Question 3
In Question 3, we are asked to solve a system of equations using the substitution method. The system of equations is given as:
Equation 1: 4x + 3y = 14
Equation 2: 2x – y = -3
We can solve this system by isolating one variable in one equation and substituting it into the other equation. Let’s start with Equation 2:
Equation 2: 2x – y = -3
Solving for y, we get: y = 2x + 3
Now we can substitute this expression for y into Equation 1:
Equation 1: 4x + 3y = 14
Substituting y = 2x + 3, we get:
4x + 3(2x + 3) = 14
Simplifying this equation, we get: 4x + 6x + 9 = 14
Combining like terms, we have: 10x + 9 = 14
Subtracting 9 from both sides, we get: 10x = 5
Dividing both sides by 10, we find: x = 0.5
Now that we have found the value of x, we can substitute it back into Equation 2 to find the value of y:
Equation 2: 2x – y = -3
Substituting x = 0.5, we get:
2(0.5) – y = -3
Simplifying this equation, we get: 1 – y = -3
Subtracting 1 from both sides, we have: -y = -4
Dividing both sides by -1, we find: y = 4
Therefore, the solution to the system of equations is x = 0.5 and y = 4.
Explanation of Question 3 Answer
Step 1: We look for two numbers that multiply to give 2 times 6 (the product of the coefficient of x^2 and the constant term) and add up to give -7 (the coefficient of x).
Step 2: The numbers that satisfy these conditions are -2 and -3. Therefore, the expression can be factored as (x – 2)(x – 3).
Step 3: To simplify further, we can expand the expression using the distributive property: x^2 – 3x – 2x + 6.
Step 4: Combining like terms, we get the simplified form of the expression as x^2 – 5x + 6.
Thus, the answer to Question 3 is x^2 – 5x + 6.
Math Nation Section 4 Test Yourself Answers: Question 4
In Question 4 of the Math Nation Section 4 Test Yourself, we are asked to solve the equation 8x + 3 = 35. To find the value of x, we need to isolate it on one side of the equation.
We start by subtracting 3 from both sides of the equation. This cancels out the 3 on the left side, leaving us with 8x = 32. Now, our equation becomes 8x = 32.
To solve for x, we divide both sides of the equation by 8. This cancels out the 8 on the left side, leaving us with x = 4. Therefore, the answer to Question 4 is x = 4.
Explanation of Question 4 Answer
The answer to question 4 in the Math Nation Section 4 test is 20. This answer is obtained by following a specific set of steps and applying the relevant mathematical concepts.
In this question, you are given the equation x + 4 = 24. The goal is to find the value of x that satisfies this equation. To do this, you need to isolate the variable x on one side of the equation.
- The first step is to subtract 4 from both sides of the equation. This cancels out the 4 on the left side, leaving you with x = 24 – 4.
- Next, you simplify the right side of the equation. 24 – 4 = 20, so the equation becomes x = 20.
Therefore, the answer to question 4 is 20. This is the value of x that satisfies the given equation x + 4 = 24.
Math Nation Section 4 Test Yourself Answers: Question 5
In question 5 of the Math Nation Section 4 Test, we are given an equation and asked to solve for the variable x. The equation is:
3x + 7 = 22
To solve for x, we need to isolate it on one side of the equation. We can do this by performing the inverse operation of addition, which is subtraction, and the inverse operation of multiplication, which is division.
First, we can start by subtracting 7 from both sides of the equation:
- 3x + 7 – 7 = 22 – 7
- 3x = 15
Next, we can divide both sides of the equation by 3 in order to solve for x:
- (3x) / 3 = 15 / 3
- x = 5
Therefore, the solution to the equation 3x + 7 = 22 is x = 5.
Explanation of Question 5 Answer
The answer to question 5 is B. Let’s review the question and the given information to understand why the answer is B.
The question states: “Which of the following expressions is equivalent to 3x – 2y + 4z?”
To find the equivalent expression, we need to simplify it by collecting like terms. The question provides three options: A) 3x + 2y – 4z, B) -2y + 3x + 4z, and C) -2y – 3x + 4z.
Since we are looking for an equivalent expression, we want to rearrange the terms so that they match the original expression. In the given expression, the terms are arranged as 3x – 2y + 4z. Option B) -2y + 3x + 4z matches this arrangement. Therefore, the answer is B.
In conclusion, the expression that is equivalent to 3x – 2y + 4z is -2y + 3x + 4z. This option rearranges the terms to match the original expression.