Identify the first step in solving this equation for x : 7(x − 8) = 49 .
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7+7
The first step in solving the equation 7(x - 8) = 49 is to distribute the 7 to the terms inside the parentheses.
The first step in solving this equation for x is to distribute the 7 to both terms inside the parentheses, which gives you: 7x - 56 = 49.
To solve the given equation 7(x - 8) = 49 for x, the first step is to distribute the 7 by multiplying it with both terms inside the parentheses. This step is known as the distributive property.
The distributive property states that for any numbers a, b, and c:
a(b + c) = ab + ac.
Applying this property to our equation, we distribute 7 to both terms inside the parentheses:
7 * x - 7 * 8 = 49.
Now we can simplify by multiplying each term:
7x - 56 = 49.
Now you have completed the first step in solving the equation.
7 + 7 is equal to 14.
Identify the second step in solving this equation for y : −3(y + 6) = 21 . (1 point)
Responses
Add 6 to both sides.
Add 6 to both sides.
Subtract 6 from both sides.
Subtract 6 from both sides.
Divide both sides by 6.
Divide both sides by 6.
Divide both sides by −3
.
The second step in solving the equation -3(y + 6) = 21 is to divide both sides of the equation by -3.
For which equation is the order of steps to first divide both sides by 6 and then to add 8 to both sides?(1 point)
Responses
8(x − 6) = 48
8 left parenthesis x minus 6 right parenthesis equals 48
6x − 8 = 48
6 x minus 8 equals 48
6(x − 8) = 48
6 left parenthesis x minus 8 right parenthesis equals 48
6(x + 8) = 48
The equation in which the order of steps is to first divide both sides by 6 and then add 8 to both sides is: 6(x - 8) = 48.
For which equation is the order of steps to first divide both sides by 2 and then to subtract 10 from both sides?(1 point)
Responses
2(x + 10) = 16
2 Left Parenthesis x plus 10 Right Parenthesis equals 16
2(x − 10) = 16
2 Left Parenthesis x minus 10 Right Parenthesis equals 16
10(x + 2) = 16
10 Left Parenthesis x plus 2 Right Parenthesis equals 16
2x + 10 = 16
2 x plus 10 equals 16
The equation in which the order of steps is to first divide both sides by 2 and then subtract 10 from both sides is: 2x + 10 = 16.
What is the solution of the equation −4(z + 12) = 36 when solving for z
? (1 point)
Responses
−21
negative 21
24
24
21
21
−6
To solve the equation -4(z + 12) = 36 for z, we will follow the steps:
1. Distribute -4 to both terms inside the parentheses:
-4z - 48 = 36
2. Add 48 to both sides to isolate the variable term:
-4z - 48 + 48 = 36 + 48
Simplifying, we get:
-4z = 84
3. Divide both sides by -4 to solve for z:
(-4z) / -4 = 84 / -4
Simplifying, we get:
z = -21
Therefore, the solution for z in the equation -4(z + 12) = 36 is -21.
Identify the first step in solving this equation for x: 7(x − 8) = 49. (1 point)
Responses
Multiply both sides by 7.
Multiply both sides by 7.
Divide only the left side by 7.
Divide only the left side by 7.
Add 8 to both sides.
Add 8 to both sides.
Divide both sides by 7.
The first step in solving the equation 7(x - 8) = 49 is to divide both sides by 7.
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In September 2019, news broke regarding progress on this 82-year-old question, thanks to prolific mathematician Terence Tao. And while the story of Tao’s breakthrough is promising, the problem isn’t fully solved yet.
A refresher on the Collatz Conjecture: It’s all about that function f(n), shown above, which takes even numbers and cuts them in half, while odd numbers get tripled and then added to 1. Take any natural number, apply f, then apply f again and again. You eventually land on 1, for every number we’ve ever checked. The Conjecture is that this is true for all natural numbers (positive integers from 1 through infinity).
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Tao’s recent work is a near-solution to the Collatz Conjecture in some subtle ways. But he most likely can’t adapt his methods to yield a complete solution to the problem, as Tao subsequently explained. So, we might be working on it for decades longer.
The Conjecture lives in the math discipline known as Dynamical Systems, or the study of situations that change over time in semi-predictable ways. It looks like a simple, innocuous question, but that’s what makes it special. Why is such a basic question so hard to answer? It serves as a benchmark for our understanding; once we solve it, then we can proceed onto much more complicated matters.
The study of dynamical systems could become more robust than anyone today could imagine. But we’ll need to solve the Collatz Conjecture for the subject to flourish.
The Collatz Conjecture is a mathematical problem that has been studied for 82 years and remains unsolved. It involves a function called f(n), which operates on natural numbers. For even numbers, it divides them by 2, and for odd numbers, it multiplies them by 3 and adds 1. The conjecture states that no matter what natural number you start with, if you repeatedly apply the function f, you will eventually reach the number 1.
In September 2019, mathematician Terence Tao made progress on the problem, but it is not yet fully solved. Tao's work provides insights into the conjecture but is not expected to lead to a complete solution. The Collatz Conjecture is a challenging problem in the field of Dynamical Systems, which studies how situations evolve over time in semi-predictable ways. Solving the conjecture would have significant implications for our understanding of dynamical systems and pave the way for more complex mathematical inquiries.
However, as of now, the full solution to the Collatz Conjecture remains elusive, and it is possible that researchers may continue working on it for many more decades.