Unit 5: Sytem of Equation & Inequalitie Homework 3: olving ytem by elimination

Answer:

6 apples and 5 bananas

Step-by-step explanation:

x + y = 11

x = 11 - y

1.75x + 0.70y = 14

1.75(11 - y) + 0.7y = 14

19.25 - 1.75y + 0.7y = 14

5.25 = 1.05y

y = 5

x = 11 - 5 = 6

Answer: 4 apples and 10 bananas

Step-by-step explanation:

Consider: a=apple, b=bananas

1) a + b = 14 // 14 total of apples and bananas

   a = 14 - b

2) 1.75a + 0.70b = $14 // $14 worth of apple and bananas

3) Substituting 1) in 2):

  1.75(14 - b) + 0.70b = 14

  24.5 - 1.75b + 0.70b = 14

   b = 10.5 / 1.05

    b = 10

4) Substituting 3) in 1):

   a = 14 - 10

   a = 4

Answer:D y=-2x-1

Step-by-step explanation:

To solve this problem, we first need to look at the direction of the line to determine whether the slope is positive or negative. Since the left portion of the line is Quadrant 2 and the right portion in quadrant 4, the slope will be negative. Anytime the line looks like \ it will be negative, while / is positive. This immediately eliminates answers A and C. Now, our next order of business is to find the slope of the line. To do this, we need to pick two points on the line and use the equation (y2-y1)/(x2-x1). Any points on the line will work, so even if you do not use the same as me, you still should get the same answer (this is a good way to check!)

point 1: (-2,3)

point 2: (0,-1)

(-1 - 3) / (0 - -2) = (-1 - 3) / (0 + 2) = -4/2 = -2

Since the slope is -2, we know the answer is d. Hope this helps

Answer: A). It is 1/6 throughout the line

Step-by-step explanation:

The amount of each trail mix that should be combined to create the special order trail mix are:

6.25 pounds of the trial that contains 75% nuts.

3.75 pounds of the trial that contains 35% nuts.

An equation is a mathematical statement that is made up of two expressions connected by an equal sign.

Example:

2x + 4 = 8 is an equation.

We have,

Trial mix A = x

= 75% nuts.

Trial mix B = y

= 35% nuts

Now,

Two equations are:

x + y = 10 _____(1)

x = 10 - y _____(2)

0.75x + 0.35y = 10 x 0.60 ______(3)

Substituting (2) in (3)

0.75 (10 - y) + 0.35y = 6

7.5 - 0.75y + 0.35y = 6

7.5 - 6 = 0.75y  - 0.35y

1.5 = 0.40y

y = 1.5/0.40

y = 3.75

Now,

x = 10 - 3.75

x = 6.25

Thus,

6.25 pounds of the trial that contains 75% nuts.

3.75 pounds of the trial that contains 35% nuts.

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The y-intercept is (0, 8).

The equation, y = mx + b, is the slope-intercept form of a straight line.

Here, x and y are the coordinates of the points, m is the gradient, and b is the intercept of the y-axis.

y = mx + b is the slope-intercept form of the equation of a straight line. In the equation y = mx + b, m is the slope of the line and b is the intercept.

The value of b is equal to y when x = 0, and m shows how steep the line is. The slope of the line is also called the gradient.

The y-intercept is what you get when you plug in x = 0

Similarly, the x-intercepts are what you get when you plug in y = 0

So, because we're finding the y-intercept, plug in 0 for x and solve for y.

Therefore, the y-intercept is (0, 8).

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Answer:

7

Step-by-step explanation:

v is is and e is 5

2v is 20 and e is 5 and plus 3

20 divided by 5 plus 3 will equal to 7

Answer:

[tex] \sf \: 2v ÷ e + 3 = 7[/tex]

Step-by-step explanation:

Given expression,

→ 2v ÷ e + 3

Now we have to use,

→ v = 10

→ e = 5

Let's solve the expression,

→ 2v ÷ e + 3

→ 2(10) ÷ (5) + 3

→ 20 ÷ 5 + 3

→ 4 + 3

→ 7 => final value

Therefore, the answer is 7.

The value of side x for the right-angle triangle is 8.6 cm.

The branch of mathematics that sets up a relationship between the sides and the angles of the right-angle triangle is termed trigonometry.

The trigonometric functions, also known as a circular, angle, or goniometric functions in mathematics, are real functions that link the angle of a right-angled triangle to the ratios of its two side lengths.

Given that the sides are Hypotenuse H = 15 cm, perpendicular = x, and the base is y.

The value will be calculated as,

x = [ sin(35) x 15 ]

x = 8.6 cm

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Answer:

The correct answer is B.

30 : 15 : 15, cookies in total, we say that y is directly proportional to x when two variables, x and y, are such that the ratio yx remains constant.

We say that y is directly proportional to x when two variables, x and y, are such that the ratio yx remains constant. If k is used to denote the constant, we can obtain the equations y = kx or y = k, where k 0.

When added to your sugar cookie batter, a teaspoon or two of extracts will significantly enhance flavor. The obvious option is vanilla. Add both vanilla and almond extracts for a more interesting flavor. Other wonderful options to think about are rum, maple, and anise.

1/30 : 2/30 : 2/30.

30 : 15 : 15.

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The company's monthly revenue rises by 8% on a monthly basis.

The mathematical formula f (x) = an x denotes an exponential function. a constant known as the function's base and a variable called x are present.

Any function whose constant is raised to the power of an input is said to be exponential.

An exponential function looks like this:

[tex]$y=a b^x$[/tex]

Where:

A stands for the starting point.

B is equal to 1 plus the rate r.

The following ordered pair can be seen in the table.

(x,y) = {(0,72000) (3,90000)}

The result is:

[tex]$90000=72000 \times b^3$[/tex]

Subtract 72000 from both sides.

[tex]$1.25=b^3$[/tex]

Take the cube's two side roots.

[tex]$b=\sqrt[3]{1.25}$$b=1.08$[/tex]

Observe that:

[tex]$b=1+r$[/tex]

The result is:

[tex]$1+r=1.08$[/tex]

Take 1 away from each side.

r= 0.08

Describe as a percentage

r= 8%

As a result, the company's monthly revenue increases by 8%.

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Slope is  $0.15 of the equation of cost  C=$0.15t+$35.00.

The slope of the line is the ratio of the rise to the run, or rise divided by the run. It describes the steepness of line in the coordinate plane.

The slope intercept form of a line is y=mx+b, where m is slope and b is the y intercept.

The slope of line passing through two points (x₁, y₁) and (x₂, y₂) is

m=y₂-y₁/x₂-x₁

Given that A cell phone company uses the equation C=$0.15t+$35.00

C is the total cost for a month of service.

t is the number of text messages.

We have to find the slope of the equation.

slope is 0.15 and 35.00 is the y intercept of the equation given.

Hence, slope is  $0.15 of the equation of cost  C=$0.15t+$35.00.

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Answer:

x = 111°

Step-by-step explanation:

Sum of interior angle

(n - 2) 180°

=(7 - 2) 180

=(5) 180°

=900°.

The value of x

=139° + 141° + 120° + x + 136° + 118° + 135° = 900°

=400° + x + 389° = 900°

=789° + x = 900°

x = 900° - 789°

x = 111°.

The side-length of square C as required in the task content is; 5 ft.

It follows from the task content that the side length of square C is to be determined according to the given diameter.

Recall that the Area, A of a square in terms of its side length, s is; A = s².

Therefore, s = √A.

On this note, for each of the given squares their side lengths are as follows;

For square A; side length is; √9 = 3ft.

For square B; side length is; √16 = 4ft.

Hence, since sides A, B and C form a right triangle;

C² = A² + B²

C² = 3² + 4²

C² = 9 + 16

C² = 25

C = 5.

Ultimately, the side-length of square C is; 5 ft.

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The distance between given pair of points ( 2, 3 ) and ( 4, 1) is equal to

2√2.

As given in the question,

Given pair of points is represented by the coordinates

( x₁ , y₁ ) = ( 2, 3 )

( x₂ , y₂ ) = ( 4, 1 )

Standard formula to find the distance between two pair of points is :

= √( y₂ - y₁ )² + (x₂ - x₁ )²

Substitute the values to get the distance between two pair of points we have,

= √ ( 1 - 3 )² + ( 4 - 2 )²

= √ 2² + 2²

= √4 + 4

= √8

= 2√2 units.

Therefore, the distance between the given pair of points is equal to 2√2.

The above question is incomplete , the complete question is:

What is the distance between the pair of points (2, 3) and (4, 1) ?

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The equation, Y = - 3x + 7 has infinitely many solutions.

Here we are given the equation y = - 3x + 7.

Now this equation has 2 variables. when an equation has 2 variables, there must be at least 2 equations for the system to have a unique solution, hence the given equation cannot have 1 unique solution.

If we plot this equation on the graph we will see that it makes a never-ending line.

Here for every value of x, we get a different value of y, which can be easily seen in the graph.

Therefore we cannot conclude that this will have no solution. Hence since there can be infinite possible answers to the above equation, this has infinitely many solutions.

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Correct Question

How many solutions does Y = - 2x + 3 have?

For the line y = -2-4x, the slope of a line parallel to this line is -4 and the slope of a line perpendicular to this line is 1/4.

What is slope?

A line's steepness and direction are measured by the line's slope. Without actually using a compass, determining the slope of lines in a coordinate plane can assist in forecasting whether the lines are parallel, perpendicular, or none at all.

The equation for the line is - y = -2 - 4x.

The slope of a line is represented by the coefficient of the x term. In this case, the slope of the line y = -2 - 4x is -4.

The slope of a line perpendicular to a given line is the negative reciprocal of the slope of the given line. The negative reciprocal of -4 is 1/4.

So, the slope of a line perpendicular to y = -2 - 4x is 1/4.

The slope of a line parallel to a given line is the same as the slope of the given line.

So the slope of a line parallel to y = -2 - 4x is -4.

Therefore, the parallel and perpendicular slope is -4 and 1/4 respectively.

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The water that the larger cup can hold more than the smaller cup, given the two cups is 15. 71 cubic inches

To find the amount of water, first find the volume of the cups using the volume of a cone formula. This formula is:

= Pi x r² x height of cone / 3

The amount of water the larger cup can hold is:

= Pi x 2² x 6 / 3

= 25. 13 cubic inches

The water in the smaller cup is:

= Pi x 1.5² x 4 / 3

= 9. 42 cubic inches

The difference is:

= 25. 13 - 9. 42

= 15. 71 cubic inches

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The coordinates of the circumcentre P(x, y) of ΔABC with vertices A (1, 0), B (8, -7), and C(7,-4) is P(6.4, 1.6).

If there is a circle that encompasses all of the vertices of a polygon, it has a circumcenter that is the centre of the circle. The circumcentre and ensuing circumcircle of a triangle are both always distinct.

Let P(x, y) be the circumcentre

Then

PA = PB = PC

⇒PA² =PB² =PC²

Now, PA² = PB²

⇒ (x−1)² +(y−0)² =(x−8)² +(y+(-7))²

⇒  x = 8 − y

And,  PB² = PC²

⇒ (x−8)² +(y+(-7))²= (x−7)² +(y−(-4))²

⇒  x = 24− 11y

Factor out x from both equations

8 − y = 24− 11y

10y = 24 - 8

10y = 16

y = 16/10

y = 1.6

For x

x = 8 − y

x = 8 − 1.6

x = 6.4

Thus, the coordinates of the circumcentre P(x, y) of ΔABC with vertices A (1, 0), B (8, -7), and C(7,-4) is P(6.4, 1.6).

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Answer:

[tex] \sf \: -4z - 5 [/tex]

Step-by-step explanation:

Given expression,

→ (-7z - 4) - (-3z + 1)

Let's simplify the expression,

→ (-7z - 4) - (-3z + 1)

→ -7z - 4 + 3z - 1

→ (-7z + 3z) + (-4 - 1)

→ (-4z) + (-5)

→ -4z - 5

Hence, the answer is -4z - 5.

[tex] \bold{(x-6)(x+6)}[/tex]

To simplify this equation, quadratic factoring patterns should be used. These patterns are the following:

[tex] \bold{a^2-b^2=(a-b)(a+b)}[/tex]

This pattern works because if one distributes the terms in the parentheses by multiplying each term in one of the parentheses by each term in the other, the result is the original equation:

[tex]\bold{36=6^2}[/tex] so it can be applied to the given equation:

[tex]\begin{gathered} \bold{x^2-36}\end{gathered}[/tex]

[tex]\bold{(x-6)(x+6)}[/tex]

∴ The factored form of the expression is (x - 6) (x + 6).

The asked quadratic equation is x²+6x+41/2 = 0

A quadratic equation is a second-order polynomial equation in a single variable x ax²+bx+c = 0. with a ≠ 0.

Given that, the roots of a quadratic equation are −6±5i

Since, -b±√(b²-4ac)/(2a) is the quadratic formula,

Therefore, we have,

b = 6, 2a = 1, a = 1/2

√(b²-4ac) = 5i

√(b²-4ac) = [tex]\sqrt{-5}[/tex]

Squaring both sides,

b²-4ac = -5

6²-4(1/2)c = -5

36-2c = -5

2c = 41

c = 41/2

Therefore, the equation = x²+6x+41/2 = 0

Hence, the asked quadratic equation is x²+6x+41/2 = 0

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To find the total number of revolutions the 32 windmills complete in 1 minute, you would use the operation of multiplication.

Multiplication is a mathematical operation used to find the total number of items in a group when the number of items in each group and the number of groups are known.

In this case, we know that there are 32 windmills and each windmill completes 120 revolutions per minute. To find the total number of revolutions completed by all 32 windmills, we need to multiply the number of windmills by the number of revolutions each windmill completes per minute.

So, the calculation would be: 32 * 120 = 3840. This calculation is read as "32 multiplied by 120 equals 3840." The result, 3840, represents the total number of revolutions completed by all 32 windmills in 1 minute.

In short, the operation of multiplication is used to find the total number of revolutions completed by all 32 windmills in 1 minute because it allows you to find the total number of items in a group when the number of items in each group and the number of groups are known.

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The five types of inequality in mathematics are greater than (>), less than (<), greater than or equal to (≥), less than or equal to (≤), and not equal to (≠). These are used to compare two values and determine if one is greater than, less than, equal to, or not equal to the other.

Greater than (>) - This type of inequality is used to compare two values and determine which one is greater than the other. For example, if we say x > y, this means that x is greater than y.

Less than (<) - This type of inequality is the opposite of the greater than inequality. If we say x < y, then this means that x is less than y.

Greater than or equal to (≥) - This type of inequality is used to compare two values and determine if one is greater than or equal to the other. This means that the two values are either equal to each other or one is greater than the other. For example, if we say x ≥ y, then this means that either x is equal to y, or x is greater than y.

Less than or equal to (≤) - This type of inequality is the opposite of the greater than or equal to inequality. If we say x ≤ y, then this means that either x is equal to y, or x is less than y.

Not equal to (≠) - This type of inequality is used to compare two values and determine if they are not equal to each other. For example, if we say x ≠ y, then this means that x is not equal to y.

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The complex number z = (4, 11π / 6) in rectangular form is z = 2√3 - i 2 (z  = 3.464 - i 2). (Correct choice: C)

Herein we find a complex number in polar form, whose definition is shown below:

[tex]z = r\cdot e^{i\,\theta}[/tex]

Likewise,

z = (r, θ)

Where:

This number is equivanlent to the following expression in rectangular form:

z = a + i b

Where:

a = r · cos θ

b = r · sin θ

If we know that r = 4 and θ = 11π / 6, then the complex number in rectangular form is:

a = 4 · cos (11π / 6)

a = 2√3

b = 4 · sin (11π / 6)

b = - 2

The rectangular form of complex number z = (4, 11π / 6) is the number z = 2√3 - i 2.

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Answer:

12x² - 40

Step-by-step explanation:

The perimeter of a two-dimensional shape is the distance all the way around the outside.

As a square has four sides of equal length, the perimeter of a square is four times the length of one side.

[tex]\begin{aligned}\textsf{Perimeter}&=4(3x^2-10)\\&=4 \cdot 3x^2+4 \cdot (-10)\\&=12x^2-40\end{aligned}[/tex]

Answer:

Perimeter of square = 12x² - 40

Step-by-step explanation:

Perimeter of square formula,

→ P = 4 × Side

→ [ P = 4a ]

Now the perimeter of square is,

→ P = 4 × (3x² - 10)

→ P = 4(3x²) - 4(10)

→ P = (4 × 3)x² - 40

→ [ P = 12x² - 40 ]

Hence, perimeter is 12x² - 40.

Answer:

x = 9

Step-by-step explanation:

first step, multiply both sides by 6. This is going to get rid of that fraction-looking situation on the left.

3x+9 / 6 = x - 3

6• 3x+9 /6 = 6(x-3)

On the left the 6s cancel (cancel is not a technical math term, but 6/6 is 1, so you are left with only the 3x+9)

3x + 9 = 6(x - 3)

Next, use distributive property on the right side.

3x + 9 = 6x - 18

Subtract 3x from both sides.

9 = 3x - 18

Add 18 to both sides.

27 = 3x

Divide both sides by 3.

27/3 = x

9 = x

x = 9

Answer: a. i. In this context, x represents the number of cups of water.

ii. In this context, y represents the number of cups of lemon juice.

iii. Yes, there is a proportional relationship between x and y. The recipe states that for every cup of water, there should be one cup of lemon juice. This means that as the amount of water increases, the amount of lemon juice also increases in the same proportion.

b. To plot the pairs in the table in a coordinate plane, we can use the values of x and y as the coordinates. For example, the first pair (x=1, y=4) would be plotted at the point (1, 4) on the coordinate plane. Similarly, the second pair (x=2, y=1) would be plotted at the point (2, 1), and so on.

Please note that the table provided with X, Y and ²14,1 is incorrect, it is not possible to plot these values in a coordinate plane as the values are not valid.

Step-by-step explanation:

The slope of 2y = -3x + 1 is  -3/2.

Given straight line is 2y = -3x + 1.

Slope of a straight line is determined by how its y-coordinate changes in respect to how its x-coordinate changes.

A line's equation can be used to determine the slope of the line.

we know that standard slope form of straight line is y = mx + c.

Where m is the slope of straight line.

Now we transform given equation to standard slope form of straight line :

Comparing this equation to standard equation, we find m = -3/2.

Therefore, the slope of given line is -3/2.

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The angle of the refracted ray is 15°.

When an incident light ray strikes the surface of the water at an angle of 20 degrees, it will be refracted, or bent, as it enters the water. The refracted ray will continue to travel through the water at a different angle than the incident ray.

In order to determine the angle of the refracted ray, we may apply Snell's law, which states that the ratio of the sines of the angles of incidence and refraction is constant for a given wave when it passes through two different media. Mathematically, this is:

n₁sin(θ₁) = n₂sin(θ₂)

Where n is the refractive index. Substituting the values given into the equation:

1.0003 * sin(20°) = 1.33 * sin(θ)

θ = 14.91

The angle of the refracted ray is 15°.

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