What is the solution to this system of linear equations?

x − 3y = −2

x + 3y = 16

Final answer:

After using the compound interest formula with an initial deposit of $25,000, an annual interest rate of 11%, and a time period of 16 years, the balance rounds to $120,034.10, which does not match any of the provided answer choices.

Explanation:

To find out how much is in the account when the child turns 16 years old, we can use the formula for compound interest: an = a1(1 + r)n, where a1 is the original amount deposited, r is the annual interest rate (expressed as a decimal), and n is the number of years the money is invested. In this case, a1 is $25,000, r is 0.11 (11%), and n is 16.

Using the formula, we calculate the account balance as follows:

Account Balance = 25,000(1 + 0.11)16

Account Balance = 25,000(1.11)16

Account Balance = 25,000(4.801364)

Account Balance = $119,999.10

However, this result is not in the given answer choices, so let's ensure we are rounding to the nearest cent:

Account Balance = $120,034.09 (before rounding)

Account Balance = $120,034.10 (after rounding to the nearest cent)

None of the answer choices matches this amount, so it is possible there has been a mistake in the provided choices or in our calculations. We should double-check the interest rate, time period, and the formula used.

To find the value of x such that the product of the first number and the cube of the second number is a maximum, we need to solve the equations and find the critical points. From the critical points, we can determine the maximum value of the product.

To find the value of x such that the product of the first number and the cube of the second number is a maximum, we need to use the given condition that the sum of twice the first number and three times the second number is 60. Let's solve this step-by-step.

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Answer:

  A.  6t - 19

Step-by-step explanation:

Put the function argument into the function definition and evaluate:

  s(2t -4) = 3(2t -4) -7 = 6t -12 -7 = 6t -19

I hope that helps!

If a point divides any line segment into two equal parts, then the point is the mid-point of that line. The ratio of the whole line segment and the distance from mid-point to any end point of the line segment is always 2:1, hence the scale factor of the dilation is 2.

To find the scale factor of the dilation, we need to find the ratio from initial point of the line segment to the distance from the point that is lie on that line segment.

Following points can be concluded from the given conditions.

The ratio of the whole line segment and the distance from mid-point to any end point of the line segment is always 2:1, hence the scale factor of the dilation is 2.

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To solve this problem, let us first assign some variables. Let us say that:

x = pigs

y = chickens

z = ducks

From the problem statement, we can formulate the following equations:

1. y + z = 30                                         ---> only chicken and ducks have feathers

2. 4 x + 2 y + 2 z = 120                      ---> pig has 4 feet, while chicken and duck has 2 each

3. 2 x + 2 y + 2 z = 90                        ---> each animal has 2 eyes only

Rewriting equation 1 in terms of y:

y = 30 – z

Plugging this in equation 2:

4 x + 2 (30 – z) + 2 z = 120

4 x + 60 – 2z + 2z = 120

4 x = 120 – 60

4 x = 60

x = 15

From the given choices, only one choice has 15 pigs. Therefore the answers are:

She has 15 pigs, 12 chickens, and 18 ducks.

Answer:

D) c - 2 = 25 + c + 10√c

Step-by-step explanation:

The given equation is sqrt(c-2) - sqrt(c) = 5

Taking square on both sides, we get

Here we used ( a+ b)^2 = a^2 + b^2 + 2ab formula.

c - 2 = 5^2 + (√c)^2 + 2(5)√c

c - 2 = 25 + c +10√c

Hope this helps!! Have a great day!! ❤

The correct equation resulting from isolating a radical term and squaring both sides of the original equation is D) [tex]\( c-2 = 25-c+10\sqrt{c} \)[/tex].

To arrive at this result, let's start with the original equation and isolate one of the radical terms:

[tex]\[ \sqrt{c-2} - \sqrt{c} = 5 \][/tex]

Now, isolate the radical on one side:

[tex]\[ \sqrt{c-2} = 5 + \sqrt{c} \][/tex]

Next, square both sides to eliminate the radical:

[tex]\[ (\sqrt{c-2})^2 = (5 + \sqrt{c})^2 \] \[ c-2 = 25 + 2 \cdot 5 \cdot \sqrt{c} + (\sqrt{c})^2 \] \[ c-2 = 25 + 10\sqrt{c} + c \][/tex]

Now, we want to isolate the term with the radical on one side and the rest on the other side:

[tex]\[ c-2 = 25 + 10\sqrt{c} + c \][/tex]

Subtract c from both sides to get:

[tex]\[ c-2-c = 25 + 10\sqrt{c} \] \[ -2 = 25 + 10\sqrt{c} \][/tex]

Finally, add 2 to both sides to isolate the radical term:

[tex]\[ -2 + 2 = 25 + 10\sqrt{c} + 2 \] \[ 0 = 25 + 10\sqrt{c} - 2 \] \[ c-2 = 25 - c + 10\sqrt{c} \][/tex]

This matches option D, confirming that the correct equation is:

[tex]\[ c-2 = 25-c+10\sqrt{c} \][/tex]

Answer:

Option: B is correct.

The factorization of the polynomial graphed below is:

f(x)=(x-2)(x-2)

Step-by-step solution:

Clearly from the graph we could see that the graph of the function touches x=2.

that means that x=2 is a root of the function

Also when the graph touches the point of x-axis and does not pass that point than that zero is the repeated zero of the function.

That means that x=2 is a repeated zero of the function f(x).

Hence,

The factorization of the polynomial graphed below is:

f(x)=(x-2)(x-2)

Hence, option B is correct.

( Also in first option:

A) x(x+2)

x=0 must also be an zero but in the graph we could see that x=0 is not a solution.

Hence option A is false.

C)

x(x-2)

again as in option: A x=0 must be a solution.

Hence, option C is false.

D)

(x+2)(x+2)

x=-2 must be a solution but the graph does not touches x=-2.

Hence, option D is incorrect )

When [tex]\(x = 6\)[/tex], the expression [tex]\(4x - 7\)[/tex] simplifies to 17 following the order of operations.

To evaluate the expression [tex]\(4x - 7\)[/tex] when [tex]\(x = 6\)[/tex], substitute the value of [tex]\(x\)[/tex] into the expression and simplify using the order of operations.

[tex]\[4x - 7\][/tex]

Replace [tex]\(x\)[/tex] with 6:

[tex]\[4(6) - 7\][/tex]

Following the order of operations (PEMDAS), perform the multiplication first:

[tex]\[24 - 7\][/tex]

Now, perform the subtraction:

[tex]\[17\][/tex]

Thus, when \(x = 6\), the value of [tex]\(4x - 7\)[/tex] is 17.

In this expression, the variable [tex]\(x\)[/tex] is multiplied by 4, and then 7 is subtracted from the result. By substituting the value of [tex]\(x\)[/tex], which is 6 in this case, and simplifying according to the order of operations, we obtain the final result of 17.

To find the number, we set up the equation 7 - 1/(3x) = 1/(2x) and solve for x.

To find the number, we need to set up an equation based on the given information.

Let's assume the number is 'x'.

According to the problem, the reciprocal of three times the number is subtracted from 7 and is equal to the reciprocal of twice the number. We can write this as:

7 - 1/(3x) = 1/(2x)

To solve this equation, we can multiply both sides by the common denominator, which is 6x. This will eliminate the fractions.

6x * 7 - 6x * 1/(3x) = 6x * 1/(2x)

42x - 2 = 3

Subtracting 2 from both sides, we get:

42x = 1

Dividing both sides by 42, we find:

x = 1/42

Therefore, the number is 1/42.

Answer:

8 half-lives of polonium-210 occur in 1104 days.

0.1174 g of polonium-210 will remain in the sample after 1104 days.

Step-by-step explanation:

Initial mass of the polonium-210 = 30 g

Half life of the sample, = [tex]t_{\frac{1}{2}}=138 days[/tex]

Formula used :

[tex]N=N_o\times e^{-\lambda t}\\\\\lambda =\frac{0.693}{t_{\frac{1}{2}}}[/tex]

where,

[tex]N_o[/tex] = initial mass of isotope

N = mass of the parent isotope left after the time, (t)

[tex]t_{\frac{1}{2}}[/tex] = half life of the isotope

[tex]\lambda[/tex] = rate constant

[tex]\lambda =\frac{0.693}{138 days}=0.005021 day^{-1}[/tex]

time ,t = 1104 dyas

[tex]N=N_o\times e^{-(\lambda )\times t}[/tex]

Now put all the given values in this formula, we get

[tex]N=30g\times e^{-0.005021 day^{-1}\times 1104 days}[/tex]

[tex]N=0.1174 g[/tex]

Number of half-lives:

[tex]N=\frac{N_o}{2^n}[/tex]

n =  Number of half lives elapsed

[tex]0.1174 g=\frac{30 g}{2^n}[/tex]

[tex]n = 7.99\approx 8[/tex]

8 half-lives of polonium-210 occur in 1104 days.

0.1174 g of polonium-210 will remain in the sample after 1104 days.

The number of unique ways that are there to arrange the letters in the word APE is 6 ways

From the question, we are to determine the number of ways the word APE can be arranged.

Since there are 3 letters in the word APE, the number of ways it can be arranged is expressed as:

3! = 3 * 2 * 1

3! = 6 ways

Hence the number of unique ways that are there to arrange the letters in the word APE is 6 ways

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Answer: 5.36

Step-by-step explanation: i got it right on my test

Using trigonometry, it can be determined that the beam will clear the wires when it stands up straight. The beam's length remains constant and by finding the height of the beam at different angles, we can confirm that it will not hit the wires.

The problem can be solved using trigonometry. Firstly, you need to find out the length of the beam when it makes an 40° angle with the ground. The length of the beam would be 8 ft / sin(40°) around 12.61 ft. Now, when the beam makes a 60° angle with the ground, the top of the beam will be sin(60°) * 12.61 ft = 10.92 ft off the ground. Because the wires are 2 ft above that (at 8 ft + 2 ft = 10 ft), the beam will clear the wires as it stands up straight.

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By applying trigonometry principles, it is determined that the beam will not clear the wires when it is lifted to stand up straight as the top of the beam at 60° angle is lower than the bottom of the wires.

To answer whether the beam will clear the wires when it is lifted, we need to apply basic trigonometry principles. First, we determine the height of the beam when it is at a 40° angle with the ground, and we know the top is 8 ft above the ground. We can use the tangent of the angle to relate this height to the length of the beam, which remains constant as the beam is raised.

So we have tan(40°) = 8ft/beam_length. Solving for beam_length, we get beam_length = 8ft/tan(40°) ≈ 9.442ft.

Next, when the beam makes a 60° angle with the ground, it is not fully raised and the wires are 2ft above the beam's top. The length of the beam when it's at this angle is beam_length = 2ft + height_at_60°. We can use the tangent function again to find this height, which gives us tan(60°) = height_at_60°/beam_length.

Solving for height_at_60°, we get height_at_60° = beam_length * tan(60°), substituting beam_length from earlier, height_at_60° = 9.442ft * tan(60°) ≈ 16.34ft.

As the bottom 2 ft of the wires are not cleared by the 16.34 ft high beam, the conclusion is that the beam will not clear the wires when it is being erected up straight.

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In order to evaluate the given expression 5a + 5b, we need to substitute the given values of a and b into the expression. The evaluated value of the expression 5a + 5b for a = -6 and b = -5 is -55.

In order to evaluate the given expression 5a + 5b, we need to substitute the given values of a and b into the expression. The value given for a is -6 and for b is -5.

So, the expression becomes: 5*(-6) + 5*(-5) = -30 - 25 = -55.

Therefore, the evaluated value of the expression 5a + 5b with a = -6 and b = -5 is -55.

The evaluated value of the expression 5a + 5b for a = -6 and b = -5 is -55.

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Answer:

x = −3, 3, −4

Step-by-step explanation: