The sum of two consecutive integers is 47. what is the product of these integers?

The correct answers are:

1. 2y=1.5x-2 ; and 4. -4y+3x=-2.

Explanation:

Using the first equation, we have the system

[tex]\left \{ {{3x-4y=2} \atop {2y=1.5x-2}} \right.[/tex]

The second equation can be written in standard form just as the first one.  To do this, we will subtract 1.5x from each side:

2y = 1.5x-2

2y-1.5x = 1.5x-2-1.5x

-1.5x+2y = -2

This makes our system

[tex]\left \{ {{3x-4y=2} \atop {-1.5x+2y=-2}} \right.[/tex]

To solve this, we will make the coefficients of x the same; we do this by multiplying the bottom equation by 2:

2(-1.5x+2y=-2)

-3x+4y=-4

This gives us the system

[tex]\left \{ {{3x-4y=2} \atop {-3x+4y=-4}} \right.[/tex]

We solve this by adding the two equations:

[tex]\left \{ {{3x-4y=2} \atop {+(-3x+4y=-4)}} \right. \\\\\\0+0 = -4\\0=-4[/tex]

There is no solution to this.

For the second system, we will follow the same process, subtract 1.5x from each side of the second equation:

2y=1.5x-1

2y-1.5x = 1.5x-1-1.5x

-1.5x+2y = -1

To make the coefficients of x the same, multiply the second equation by 2:

2(-1.5x+2y=-1)

-3x+4y=-2

We will add the two equations to solve:

[tex]\left \{ {{3x-4y=2} \atop {+(-3x+4y=-2)}} \right. \\0+0=0\\0=0[/tex]

This means that the equations are of the same line and there are infinite solutions.

For the third system, the coefficients are already the same.  We will cancel by subtracting the bottom equation from the top:

[tex]\left \{ {{3x-4y=2} \atop {-(3x+4y=2)}} \right. \\\\-4y-4y=2-2\\-8y=0\\\frac{-8y}{-8}=\frac{0}{-8}\\y=0[/tex]

Since we have a value for y, this has a solution.

For the last system, we will rearrange the second equation with the x term in front:

3x-4y=-2

Now we will subtract this from the first equation:

[tex]\left \{ {{3x-4y=2} \atop {-(3x-4y=-2)}} \right. \\-4y--4y=2--2\\0=4[/tex]

This has no solution.

The phrase 'three times a number plus 16' translates to '3x + 16' in mathematical terms, where 'x' represents any number.

The phrase 'three times a number plus 16' can be converted into an algebraic expression. In mathematical terms, 'a number' is usually represented by the letter 'x'. 'Three times a number' can be written as '3x'. 'Plus' is represented by the '+', so 'three times a number plus 16' is written as '3x + 16'.

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Answer: Set goals higher than what you feel you can achieve.

Step-by-step explanation:

Answer:

The y-coordinate of the solution is approximately 7.2.

Explanation:

To solve the system of equations:

1. [tex]\( 5x + 2y = 7 \)[/tex]

2. [tex]\( -2x + 6 = 9 \)[/tex]

We'll start by solving the second equation for [tex]\( x \):[/tex]

[tex]\[ -2x + 6 = 9 \][/tex]

Subtract 6 from both sides:

[tex]\[ -2x = 9 - 6 \][/tex]

[tex]\[ -2x = 3 \][/tex]

Divide both sides by -2:

[tex]\[ x = -\frac{3}{2} \][/tex]

Now that we have the value of [tex]\( x \)[/tex], we can substitute it into the first equation to find [tex]\( y \):[/tex]

[tex]\[ 5x + 2y = 7 \][/tex]

[tex]\[ 5(-\frac{3}{2}) + 2y = 7 \][/tex]

[tex]\[ -\frac{15}{2} + 2y = 7 \][/tex]

Add [tex]\( \frac{15}{2} \)[/tex] to both sides:

[tex]\[ 2y = 7 + \frac{15}{2} \][/tex]

[tex]\[ 2y = \frac{14}{2} + \frac{15}{2} \][/tex]

[tex]\[ 2y = \frac{29}{2} \][/tex]

Divide both sides by 2:

[tex]\[ y = \frac{29}{4} \][/tex]

Now, let's convert the fraction into a decimal rounded to the nearest tenth:

[tex]\[ y \approx 7.2 \][/tex]

At a value of 2, the graph of F(x) = 2/x-2 has a vertical asymptote.

Final answer:

The rate of change of the surface area of a sphere with respect to its radius is 8πr. For a sphere with a radius of 6, this rate is 48π square units per unit of radius.

Explanation:

The surface area S of a sphere is given by the formula S = 4πr², where r is the radius of the sphere. To find the rate of change of the surface area with respect to the radius, we need to differentiate the surface area formula with respect to r. This calculation gives us dS/dr = 8πr. When r = 6, the rate of change, or the derivative evaluated at this radius, is dS/dr = 8π(6) which equals 48π.

So, the rate of change of the surface area of a sphere with respect to its radius when r = 6 is 48π square units per unit of radius.

The three consecutive even integers are 10, 12, and 14.

Given that three consecutive even number the sum of the smallest integer and twice the median is 20 more than the largest integer.

Let's represent the three consecutive even integers as follows:

The smallest integer: x

The median integer: x + 2 (since it's consecutive and even)

The largest integer: x + 4 (since it's consecutive and even)

Now, we can set up the equation based on the given information:

"The sum of the smallest integer and twice the median is 20 more than the largest integer."

This can be expressed as:

x + 2(x + 2) = (x + 4) + 20

Now, we can solve for x:

x + 2x + 4 = x + 24

Combine like terms:

3x + 4 = x + 24

Subtract x from both sides:

2x + 4 = 24

Subtract 4 from both sides:

2x = 20

Now, divide by 2:

x = 10

So, the smallest integer is 10, the median integer is x + 2 = 12, and the largest integer is x + 4 = 14.

Therefore, the three consecutive even integers are 10, 12, and 14.

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Final answer:

The notation f'(x) represents the derivative of the function f with respect to x, signifying the rate of change of the function or the slope of the tangent to its curve at a specific point.

Explanation:

The notation f prime of x, often written as f'(x), represents the derivative of the function f with respect to x. This concept is a fundamental part of calculus, which deals with the rate at which quantities change. The derivative f'(x) is the instant rate of change of the function f(x) at a particular value of x, or the slope of the tangent line to the curve f(x) at that point.

For example, if f(x) = x2, then the derivative f'(x) would equal 2x, as the rate of change of x2 with respect to x is 2x. This means that for every unit increase in x, the value of x2 increases by an amount corresponding to 2x.

The derivative can also be understood in the context of physics, where it might represent things like velocity, which is the derivative of position with respect to time. Regardless of the context, f'(x) is about understanding how a function changes at a specific point.

24 -18 = 6

24/6 =4

18/6 =3

6 bundles with 4 balls and 3 clubs each

Final answer:

The height of the cone is equal to the slant height plus 1.

Explanation:

To find the height of the cone, we can use the formula for the surface area of a cone: S = πr^2 + πrl, where r is the radius of the base and l is the slant height. Since the problem states that the surface area is equal to the volume, we can set S equal to V and solve for h, the height of the cone.

Here's the step-by-step calculation:

S = V

πr² + πrl = V

πr² + πrl = πr²h

πr² + πrl - πr²h = 0

Combine like terms: πr²(1 - h) + πrl = 0

Divide both sides by πr to solve for h: 1 - h + l = 0

Subtract l from both sides: 1 - h = -l

Subtract 1 from both sides: -h = -l - 1

Multiply both sides by -1 to solve for h: h = l + 1

So, the height of the cone is h = l + 1.

Food, clothing, and medical are variable expenses because they occur regularly but can change from month to month.

The Porters’ food, clothing, and medical expenses are variable expenses.

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0.075 m is 7.5 centimeters.

A unit conversion expresses the same property as a different unit of measurement.

Given that, convert 0.075 m into cm

We know that, 1 meter = 100 centimeters

0.075 meter = [(100) × 0.075] centimeter

0.075 meter = (100 × 0.075) centimeter

= (100 × 75/1000) centimeter

= (7500/1000) centimeter

= (75/10) centimeter

= 7.5 centimeter.

Hence, 0.075 m is 7.5 centimeters.

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The given difference of two numbers which we have to estimate is

  920,026-535,722

Actual Difference=384,304

→Estimation Using Ones

 920,030-535,720=384,310

→Estimation Using Tens

 920,000-535,700=384,300

Estimation using Tens is Closer to Actual Difference.

So, Answer=384,300

To find the value of k when x^2+kx+7 is equivalent to (x+1)(x+h), we expand (x+1)(x+h), compare coefficients, and deduce that k is 8.

If h and k are constants and x^2+kx+7 is equivalent to (x+1)(x+h), we are looking for the value of k. To find k, we expand the right-hand side and then compare coefficients.

Expanding (x+1)(x+h) gives us x^2+hx+x+h, which simplifies to x^2+(h+1)x+h.

Comparing this to x^2+kx+7, we can see that k must be equal to h+1.

Since there is no h term in x^2+kx+7, we infer that h must be equal to 7. Consequently, k=h+1 which leads us to k=7+1.

Therefore, the value of k is 8.