Solve each system of equations by graphing. Identify all possible solutions.
y=x²+6x
y=x-4

In order to show that △ ABC ≅ △ DEF by SAS congruence, we need the sides of triangles ABC and DEF with the angle between the respective sides.

Triangle congruence: Two triangles are said to be congruent if all three of their corresponding sides and all three of their corresponding angles have the same measurements. Slide, twist, flip, and turn these triangles to create an identical appearance. They are in alignment with one another when moved. "" is the congruence symbol. When two figures are comparable to one another in terms of their size and shape, this is what mathematicians mean by congruence. In essence, there are four congruence principles that demonstrate if two triangles are congruent. However, locating all six dimensions is essential. As a result, the congruence of triangles can be assessed using just three of the six variables.

Congruency Requirements:

SSS (Side-Side-Side)

SAS (Side-Angle-Side)

ASA (Angle-Side-Angle)

AAS (Angle-Angle-Side)

RHS (Right angle-Hypotenuse-Side) 

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Answer:

c=13.4

Step-by-step explanation:

Step-by-step explanation:

2×length + 2×width = 204 cm

length = x

width = 3×(x + 2)

as consecutive even integers are always spaced out by 2 (2, 4, 6, ...).

we are using these 2 equations in the first main equation :

2×x + 2×3×(x + 2) = 204

2x + 6x + 12 = 204

8x + 12 = 204

8x = 192

x = 192/8 = 24

length = x = 24 cm

width = 3(x + 2) = 3×26 = 78 cm

Answer:

576 ft squared

Step-by-step explanation:

8 x 72 ft= 576 total area

Hope this helps!

The range of a graph can be found using the values depicted on the y-axis.

According to the question,

We have the following information:

We have a graph and we need to know how do we find the range of that graph.

We know that the domain of a graph is the collection or in other words set of input values depicted on x -axis and the range of a graph is the set of possible output values portrayed on y-axis of the graph.

In noting down the range of a graph, we first list down the down value and then the up value or in other words we move from down to up in order to find the range of a graph.

Hence, the range of a graph can be found using the values depicted on the y-axis.

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The variable in this expression 4b + 2b + 1⁄4 is b

From the question. we have the following expression

4b + 2b + 1⁄4

Take for instance, we have an expression

Ax

Where A is a number

It means that the expression x is a variable

This is so because the value of x can change

Similarly, the variable in 4b + 2b + 1⁄4 is b

This is so because the value of b can change

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The relation:

(9, -1) (4,2) (5,3) (8, -1) (3, 0)

Is a function, because each element of the domain appears only once.

A relation maps elements from a set, called the domain, into elements from another set, called the range.

Such that the element x being mapped into the element y is written as:

(x, y).

A relation is a function only if each element in the domain is mapped into only one element of the range.

Here we have the relation:

{  (9, -1) (4,2) (5,3) (8, -1) (3, 0)}

We can see that no element of the domain appears twice, so yes, this is a function.

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Using the formula of simple interest and concept of percentage, the amount earned per year is $9542 and the amount spent on freight is $6396

Simple interest is calculated with the following formula: S.I. = P × R × T, where P = Principal, R = Rate of Interest in % per annum, and T = Time, usually calculated as the number of years. The rate of interest is in percentage r% and is to be written as r/100.

1) To solve the amount earned every year, we have to use the formula of simple interest here;

S.I = 73400(1 + 0.13*1)

S.I = 82,942.00

The interest earned per year will be

82942 - 73400 = 9,542.00

2) Using percentage concept here

x = 0.06 * 106600

x = 6396

The amount spent on freight is $6396

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The inequality is equivalent to 5+y+b0.

A positive number divided by both sides of an inequality results in an equal inequality. And if the inequality symbol is reversed, division on both sides of an inequality with a negative value results in an analogous inequality.

Inequalities that convey the same information are referred to as equivalent inequalities.

In mathematics, larger than symbol (>), less than symbol (), less than or equal to symbol (), and not equal to symbol () are the five inequality symbols.

Algebraic equations with equivalent solutions or roots are called equivalent equations. An analogous equation is one that is created by adding or removing the identical quantity or expression from both sides of a given equation.

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The multiples of a number are numbers found by multiplying the original number by any whole number. Prime multiples are number which multiply each other and form a prime

What are Prime Numbers ?

A prime number is a number that is only divisible by 1 and itself. So, prime numbers have only two factors, 1 and the number itself.

This means these numbers cannot be divided by any number other than 1 and the number itself without leaving a remainder.

Numbers that have more than 2 factors are known as composite numbers.

For example :

The factors of 3 are 1 and 3.

Therefore, the prime multiples of 3 are 1 and  3.

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What is division?

Division is the process by which value of single unit can be calculated from the value of multiple unit.

The number to be divided is known as dividend, the number by which the dividend is divided is the divisor, the result obtained is the quotient and the remaining part is the remainder.

There is a well known formula for division

Divisor x Quotient + Remainder = Dividend.

This is a problem sum of division

Cost of 15 flowers = $200

Cost of 1 flower = $200 [tex]\div[/tex] 15 = $13.33

Cost of 5 flowers = $70

Cost of 1 flower = $70 [tex]\div[/tex] 5 = $14

So unit price of 15 flower arrangement of $200 is lower than unit price of 5 flower arrangement of $70

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The area of triangle ABC is 18cm².

A triangle is a three-sided polygon that consists of three edges and three vertices. The most important property of a triangle is that the sum of the internal angles of a triangle is equal to 180 degrees.

This property is called angle sum property of a triangle. A triangle is a polygon with three edges and three vertices. It is one of the basic shapes in geometry.

= 72*sin(75)*cos(75)

= 36*(2sin(75)cos(75))

= 36*sin(150)

=  18

In conclusion, the area of the triangle is 18cm².

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Answer:

To find a polynomial, you must identify the terms, coefficients, and degrees. The degree of a polynomial is the highest exponent of the variable in the equation. The coefficients of a polynomial are the values that are multiplied by the variables. The terms of a polynomial are the pieces of the equation, including the coefficients and the variables. After identifying the terms, coefficients, and degree of the polynomial, you can write it in standard form.

Step-by-step explanation:

Answer:

-2

Step-by-step explanation:

a is -3

d is -1

you subtract the two

-3 - (-1)

when you have two negatives they cancel each other out and become positive

-3 + 1

-3 + 1 = -2

that is your answer

The author gives a detailed description of a healthy ecology D. To contrast how the town changed once the ecology was compromised with the use of harmful pesticides.

What is the ecology about?

Ecology also teaches us about the benefits of ecosystems and how to use the Earth's resources in a way that protects the environment for future generations. We can learn more about our surroundings thanks to the various fields of ecology, which include statistical ecology, marine ecology, and vegetation ecology.

With the help of this information, we can manage our natural resources, protect human health, and improve the environment. In this case, the author give a detailed description of a healthy ecology.

The correct option is D.

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Answer:

A)  61.93 minutes

B)  40.93 minutes

Step-by-step explanation:

Newton's law of cooling

[tex]T(t)=T_S+(T_0-T_S)e^{-kt}[/tex]

where:

Given values:

Substitute the given values into the formula and solve for k:

[tex]\implies 60=20+(90-20)e^{-25k}[/tex]

[tex]\implies 40=70e^{-25k}[/tex]

[tex]\implies e^{-25k}=\dfrac{4}{7}[/tex]

[tex]\implies \ln e^{-25k}=\ln \dfrac{4}{7}[/tex]

[tex]\implies -25k \ln e=\ln \dfrac{4}{7}[/tex]

[tex]\implies -25k=\ln \dfrac{4}{7}[/tex]

[tex]\implies k=-\dfrac{1}{25}\ln \dfrac{4}{7}[/tex]

Given values:

Substitute the given values into the formula and solve for t:

[tex]\implies 30=20+(60-20)e^{\frac{1}{25}t\ln \frac{4}{7}}[/tex]

[tex]\implies 10=40e^{\frac{1}{25}t\ln \frac{4}{7}}[/tex]

[tex]\implies e^{\frac{1}{25}t\ln \frac{4}{7}}=\dfrac{1}{4}[/tex]

[tex]\implies \ln e^{{\frac{1}{25}t\ln \frac{4}{7}}}=\ln \dfrac{1}{4}[/tex]

[tex]\implies {\dfrac{1}{25}t\ln \dfrac{4}{7}}\ln e=\ln \dfrac{1}{4}[/tex]

[tex]\implies {\dfrac{1}{25}t\ln \dfrac{4}{7}}=\ln \dfrac{1}{4}[/tex]

[tex]\implies t=\dfrac{\ln \dfrac{1}{4}}{{\frac{1}{25}\ln \frac{4}{7}}}[/tex]

[tex]\implies t=61.93063129...[/tex]

[tex]\implies t=61.93\;\; \sf minutes[/tex]

Given values:

Substitute the given values into the formula and solve for t:

[tex]\implies 30=-10+(90-(-10))e^{\frac{1}{25}t\ln \frac{4}{7}}[/tex]

[tex]\implies 40=100e^{\frac{1}{25}t\ln \frac{4}{7}}[/tex]

[tex]\implies e^{\frac{1}{25}t\ln \frac{4}{7}}=0.4[/tex]

[tex]\implies \ln e^{\frac{1}{25}t\ln \frac{4}{7}}=\ln 0.4[/tex]

[tex]\implies{\dfrac{1}{25}t\ln \dfrac{4}{7}}\ln e=\ln 0.4[/tex]

[tex]\implies {\dfrac{1}{25}t\ln \dfrac{4}{7}}=\ln 0.4[/tex]

[tex]\implies t=\dfrac{\ln 0.4}{{\frac{1}{25}\ln \frac{4}{7}}}[/tex]

[tex]\implies t=40.93392072...[/tex]

[tex]\implies t=40.93\;\; \sf minutes[/tex]

Answer:

At least 8 rows

Step-by-step explanation:

45 divided by 9 is 5. That is 5 rows

9x3= 27 and there are 23 adults. So that is 3 rows

5+3=8

8 rows in total.

The function f(x) is represented by the graph attached below. The domain of the graph is given by [-5,5] or {x | -5 ≤ x ≤ 5 ,x ∈ R} .

A function's domain in mathematics is the set of potential inputs that it can take. It can also be written as b, where f is the function.

Formally speaking, X is the domain of f given a function. It is important to note that, in terms of contemporary mathematics, a function's domain is a part of its definition rather than a property.

The function f can be graphed in the Cartesian coordinate system in the uncommon case where X and Y are both subsets of. In this instance, the function's graph, which is projected onto the graph's x-axis, serves as a representation of the domain.

A real function, f, may not be defined for all possible values of the variable if it is specified in a formula. The natural domain, also known as the domain de definition of f, is the set of real numbers to which the formula in this case can be evaluated to either a real number. It is common to refer to a partial function as just a function and its natural domain as just a domain.

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The used postulate is explained below.

What is congruent triangles?

Two triangles are stated to be congruent if the 3 facets and the 3 angles of each the angles are same in any orientation.

When angles and facet blanketed in among those angles of 1 triangle are identical as any other one, each those triangle are Congruent.

This standards of congruency is is aware of as ASA criterion as a facet blanketed among angles are all identical.

Two triangles are congruent, if angles and the facet blanketed among them in a single triangle is same to the 2 angles and the facet blanketed among them of the alternative triangle.

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The value of the variables x and y are,

11) x = 13,  y = 18.4

12) x = 6.9,  y = 8.

What is right angle triangle?

A right triangle is a triangle with one angle at a right angle, meaning that two of its sides are perpendicular. It is also known as a right-angled triangle, an orthogonal triangle, or more commonly as a rectangled triangle. Trigonometry's foundation is the right triangle's relationship between its sides and other angles.

11)Let the given triangle is a right angle triangle.

So we can use trigonometric ratio,

[tex]tan(45) = \frac{x}{13} \\x = 13(tan(45))\\x = 13[/tex]

Using this to find the value of y.

[tex]sin(45) = \frac{x}{y} \\sin(45) = \frac{13}{y} \\y = \frac{13}{sin(45)} \\y = 18.4[/tex]

So, the values of x and y are x = 13 and y = 18.4.

12) The given triangle is a right angle triangle.

So we can use trigonometric ratio,

[tex]tan(30) = \frac{4}{y}\\ y = \frac{4}{tan(30)} \\y = 6.9[/tex]

Now to find the value of x.

[tex]sin(30) = \frac{4}{x} \\x = \frac{4}{sin(30)} \\x = 8[/tex]

Hence, the values of x and y are, x = 6.9 and y = 8.

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The slope of the perpendicular line to the equation 5x - 4y = -24, is: 4/5.

The slope values of two lines which lie perpendicular to each other on a coordinate plane are always negative reciprocals to each other.

Rewrite the equation, 5x - 4y = -24 in slope-intercept form:

5x - 4y = -24

-4y = 5x - 24

Divide both sides by -4

-4y/-4 = 5x/-4 - 24/-4

y = -5/4x + 6

The slope of y = -5/4x + 6 is -5/4. The negative reciprocal of -5/4 is 4/5. Thus, the slope of the line that is perpendicular to the equation of the given line is: 4/5.

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The equation that best represents the area of the forest each year is A. y = 2,000(0.94)^x.

An equation is the statement that illustrates the variables given. In this case, two or more components are taken into consideration to describe the scenario.

Since the certain forest covers an area of 2,000 square kilometers and each year, this area decreases by 6%.

The equation will be:

= 2000(1 - 6%)^x

= 2000(0.94)^x

In conclusion, the correct option is A.

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The shape of Keisha's rectangular shaped planter would be the shape with the dimensions shown in the image below.

The volume of a rectangular box = length × width × height.

The shape of the planter is a rectangular box.

Given that the rectangular shaped planter would be filled first by 4 in. of rocks, it means the height of the rectangular shaped planter that the rock will occupy up to would be 4 in.

Find the volume the rock would fill:

Volume occupied by the rocks in the rectangular shaped planter = (10)(10)(4) = 400 in.²

If the soil to be filled must be exactly 800 in.², then the total volume of the shape of the rectangular planter that would be used would be: 400 + 800 = 1,200 in.².

From the options given, the last shape that has a height of 12 in., length of 10 in, and width of 10 in will have a volume = (12)(10)(10) = 1,200 in.².

Therefore, the last shape in the diagram below could be the shape of Keisha's rectangular shaped planter.

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The area of the rectangular paper left after two semi circles are cut out of it is 186.96 cm².

A rectangle is a closed 2-D shape, having 4 sides, 4 corners, and 4 right angles (90°).The opposite sides of a rectangle are equal and parallel.

To caluculate the area of the paper that remains, we use the formula below.

Formula:

Where:

From the question,

Given:

Substitute these values into equation 1

Hence, the area of the paper left is 186.96 cm².

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Answer: If two parallel lines are cut by a transversal, then corresponding angles are congruent.

Step-by-step explanation:

The perimeter of triangle CED is 58 in

What is a Parallelogram ?

A unique variety of quadrilateral called a parallelogram has both pairs of its opposite sides parallel and equal.

Parallelograms' characteristics

(1) The opposing sides of a parallelogram are parallel to one another. In this case, AB || CD and AC || BD

(2) A parallelogram has equal-length opposing sides. In this case, AB = CD and AC = BD

(3) In a parallelogram, the measurements of the opposing angles are equal. In this case, A Equals C and B = D.

(4) The total of all the angles of a parallelogram, like all other quadrilaterals, is 360°.

(5) A parallelogram has neighboring or contiguous angles that sum to 180 degrees. As a result, A + B, B + C, C + D, and D + A all equal 180 degrees.

(6) A parallelogram's diagonals cut each other in half. In this case, OA = OC and OB = OD

(7) The parallelogram in the figure is split into two congruent triangles by the diagonals AC and BD.

The diagonals of a parallelogram bisect each other AC = 2 CE

Substitute AC= 50 into AC = 2CE = 50 = 2CE

Calculate 50 = 2 CE=>  CE = 25

The diagonals of a parallelogram bisect each other BD = 2DE

Substitute BD= 30 into BD = 2DE =>  30 = 2DE

Calculate 30 = 2DE  => DE = 15

Opposite sides of parallelogram are congruent AB = CD

Substitute AB = 18 into AB = CD =>  CD = 18

Perimeter of a triangle PΔCDE = CE + CD + DE

Substitute  CE = 25, DE = 15, CD = 18.

PΔCDE = CE + CD + DE

PΔCDE = 25+ 18 + 15

PΔCDE = 58 in

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The amount of time it will take both of them is 5.5 days, if Ann and Ben work together

The given parameters are

Ann = 10 days

Ben = 12 days

Represent the number of days it takes both of them with T

So, we have the following equation

1/T = 1/Ann + 1/Ben

Substitute the known values in the above equation, so, we have the following representation

1/T = 1/10 + 1/12

So, we have

1/T = (12 + 10)/120

1/T = 22/120

Take the reciprocal of both sides

T = 120/22

Evaluate

T = 5.5

Hence, it will take them 5.5 days

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If triangle has 2 congruent sides and 2 congruent angles ,such a triangle is known as Isosceles triangle.

What is congruent triangle?

Congruence of triangles: 2 triangles square {measure} aforesaid to be congruent if all 3 corresponding sides square measure equal and every one the 3 corresponding angles square measure equal in measure. These triangles may be slides, rotated, flipped and turned to be looked identical. If repositioned, they coincide with one another. The image of congruity is’ ≅’.

Main Body:

A triangle with two congruent sides and congruent base angles is termed as an isosceles triangle.

If two sides of a triangle are congruent, then by isosceles triangle property, the angles opposite to these sides are congruent.

That is, base angles of the triangle are congruent.

Then, such a triangle is known as Isosceles triangle.

Therefore, option B is correct.

The actual question with option is in figure.

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Total 56 pencils are needed for the exam.

Multiplication is one of the arithmetic operation used to find total numbers of items or things.

Given,

Total students in the classroom = 28

Total number of pencils needed by each student for the exam=2 pencils

Total number of pencils needed for the exam are,

= Total student ×  Number of pencils needed by each student

= 28 × 2

= 56 pencils

Hence , The total number of pencils needed by the students for the exam would be 56 .

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Answer:

c = 5m + 30

Step-by-step explanation:

It is given that:

c = total amount of chickens.

m = total amount of months.

x = amount of chickens per month.

30 is the base amount that Junior will have no matter how much months have passed.

Plug in 4 for m (months), and 50 for c (total amount of chicken) in the given equation:

c = xm + 30

50 = x(4) + 30

50 = 4x + 30

Isolate the variable, x. Note the equal sign, what you do to one side, you do to the other. Do the opposite of PEMDAS. PEMDAS is the order of operations, and stands for:

Parenthesis

Exponent (& root)

Multiplication

Division

Addition

Subtraction

~

First, subtract 30 from both sides of the equation:

50 = 4x + 30

50 (-30) = 4x + 30 (-30)

50 - 30 = 4x

20 = 4x

Next, divide 4 from both sides of the equation:

(20)/4 = (4x)/4

x = 20/4

x = 5

The amount of chicken per month is 5.

Set the equation. Plug in 5 for x in the given equation:

c = xm + 30

c = 5m + 30

c = 5m + 30 is your answer.

Check. Plug in time per month and find the number of chickens:

Time in months = 0 ; Number of Chickens = 30

(30) = 5(0) + 30

30 = 0 + 30

30 = 30

Time in months = 4 ; Number of chickens = 50

(50) = 5(4) + 30

50 = 20 + 30

50 = 50

Time in months = 720 ; Number of chickens 3630

(3630) = 5(720) + 30

3630 = 3600 + 30

3630 = 3630

~