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Q1: How do you identify the solution to a system of two linear equations in two variables from their graphs?
Q2: What causes a system of two equations in two variables to have no solution? What causes a system of two equations in two variables to have infinitely many solutions?
Q3: How do you determine whether to use substitution or elimination to solve a system of equations?
Q1: The solution of a system of two linear equations in two variables is the point where their graphs intersect. At that point, the (x, y) pair satisfies both equations.
__
Q2: There will be no solution when the lines representing the equations do not intersect. That will only happen when they are parallel. (We say the equations are "inconsistent.")
There will be infinitely many solutions when the lines overlap. That is, the equations for them are essentially the same equation. (We say the equations are "dependent.")
__
Q3: Elimination is easiest when the coefficients of one of the variables are opposites of each other, or if one is a factor of the other. Substitution is easiest if one of the equations is easily put in the form x = ( ) or y = ( ).
If neither or both conditions hold, the choice is by student preference. (In this latter case, solution using Cramer's rule may be simplest.)
Need help with a math question
ACD and ACE are supplementary angles that need to equal 180 degrees
ACE = 180 -100 = 80 degrees.
For this case we have by definition that, a flat angle is the space included in an intersection between two straight lines whose opening measures 180 degrees. So, the angle between A and B is 180 degrees.
If we are told that the ACD angle is 100 degrees, then the DCB angle should be 80 degrees.
The angle ACE is opposite by vertex to the angle DCB, then it is also 80 degrees; since by definition, two angles opposed by the vertex are congruent or equal.
Answer:
80 degrees
Find the area of a parallelogram if a base and corresponding altitude have the indicated lengths. Base 3.5 feet, altitude 3/4 feet. 2 1/8 sq. Ft. 2 5/8 sq. Ft. 3 3/8 sq. Ft.
A = bh
A = (3.5)((0.75)
A = 2.625
A = 2_5/8 ft^2
How do you find the length of the legs of an isosceles triangle?
Answer:
Step-by-step explanation:
Choose the system of equations which matches the following graph:
a line includes points 0 commas 2 and 5 commas 0
A) 2x − 5y = 10
4x − 10y = 20
B) 2x + 5y = 10
4x + 10y = 20
C) 2x + 5y = 10
4x − 10y = 20
D) 2x − 5y = 10
4x + 10y = 20
Answer:
B
Step-by-step explanation:
If you put (x,y) values in here
(0,2)
2x + 5y = 10
2.0 + 5.2 = 10
0 + 10 = 10
4x + 10y = 20
4.0 + 10.2 = 20
0 + 20 = 20
And the other is
(5,0)
2x + 5y = 10
2.5 + 5.0 = 10
10+0= 10
4x + 10y = 20
4.5 + 10.0 = 20
20 + 0 = 20
All of them is OK.
Answer:
2x + 5y = 10
4x + 10y = 20
Step-by-step explanation:
just took the test
Please explain how to solve step by step. Thank you.
3x-2 = 5x-14
Subtract 3x from each side:
-2 = 2x -14
Add 14 to each side:
12 = 2x
Divide both sides by 2:
x = 12/2
x = 6
How many possible rational roots are there for 2x4+4x3−6x2+15x−12=0?
A. 4
B. 6
C. 8
D. 12
E. 16
Final answer:
Using the Rational Root Theorem, the polynomial equation 2x⁴+ 4x³-6x²+15x-12=0 has 12 distinct possible rational roots, where factors of the constant term are divided by factors of the leading coefficient.
Explanation:
To determine the number of possible rational roots for the polynomial equation 2x⁴+4x³−6x²+15x−12=0, we can use the Rational Root Theorem. This theorem states that if the polynomial has any rational roots, they must be of the form ±p/q, where p is a factor of the constant term and q is a factor of the leading coefficient.
The constant term in this case is -12, and its factors are ±1, ±2, ±3, ±4, ±6, and ±12. The leading coefficient is 2, and its factors are ±1 and ±2.
Thus, the possible rational roots are combinations of the factors of -12 divided by the factors of 2, giving us ±1, ±2, ±3, ±4, ±6, and ±12 combined with ±1 and ±2. So, our possible rational roots are: ±1/1, ±2/1, ±3/1, ±4/1, ±6/1, ±12/1, ±1/2, ±2/2, ±3/2, ±4/2, ±6/2, and ±12/2 which simplifies to ±1/1, ±2, ±3, ±4, ±6, ±12, ±1/2, ±3/2, and ±6/2.
After simplification, there is a total of 12 distinct possible rational roots, corresponding to option D.
Which of the following quadrilaterals have diagonals that are perpendicular to each other? Check all that apply.
Answer: Option B and option D.
Step-by-step explanation:
We know that a quadrilateral is a 2-dimensional closed shape that has four sides.
By definition the diagonals of a quadrilateral are the lines that connect two non-adjacent vertices.
The following quadrilaterals have diagonals that are perpendicular to each other (also known as perpendicular bisector diagonals), which means that they form four angles of 90 degrees (right angles): Rhombus and Square.
Therefore the answers are: the option B and the option D.
The quadrilaterals that have diagonals perpendicular to each other are:
a. Rectangle
b. Rhombus
d. Square
In a rectangle, the diagonals are always perpendicular to each other. The same applies to a rhombus, which is a special case of a parallelogram with all sides equal.
Similarly, a square is a special type of rectangle and rhombus, so its diagonals are also perpendicular. However, in a general parallelogram, the diagonals are not necessarily perpendicular. Therefore, the correct options are a, b, and d.
Therefore, The quadrilaterals that have diagonals perpendicular to each other are:
a. Rectangle
b. Rhombus
d. Square
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A rain gutter is to be constructed from a metal sheet of width 30 cm by bending up one-third of the sheet on each side through an angle θ. How should θ be chosen so that the gutter will carry the maximum amount of water?
Answer:
θ = 60°
Step-by-step explanation:
The cross sectional area of the trapezoid shape will be that of a trapezoid with bases of 10 cm and (10 cm + 2·(10 cm)·cos(θ)) and height (10 cm)·sin(θ).
That area in cm² is ...
A = (1/2)(b1 +b2)h = (1/2)(10 + (10 +20cos(θ))(10sin(θ)
A = 100sin(θ)(1 +cos(θ))
A graphing calculator shows this area to be maximized when ...
θ = π/3 radians = 60°
_____
A will be maximized when its derivative with respect to θ is zero. That derivative can be found to be 2cos(θ)² +cos(θ) -1, so the solution reduces to ...
cos(θ) = 1/2
θ = arccos(1/2) = π/3
The recyclables have been sorted into plastic and glass. The number of plastic items is 30 less than nine times the number of glass items. The recyclables are 85% plastic. How many glass items are there?
a. 6
b. 9
c. 15
d. 18
Answer:
B
Step-by-step explanation:
We will use substitution method after writing 2 equations to solve for number of glass items.
Let number of glass items be g and number of plastic items be p
"The number of plastic items is 30 less than nine times the number of glass items":
We can write the equation as [tex]p=9g-30[/tex]
"The recyclables are 85% plastic":
This means that plastic (p) is 0.85 of the total (p+g). Thus we can write equation as [tex]p=0.85(p+g)[/tex]
We can substitute the expression for p in first equation into 2nd and solve for g:
[tex]p=0.85(p+g)\\(9g-30)=0.85((9g-30)+g)\\9g-30=0.85(10g-30)\\9g-30=8.5g-25.5\\0.5g=-25.5+30\\0.5g=4.5\\g=\frac{4.5}{0.5}\\g=9[/tex]
The number of glass items is 9
Answer:
b. 9
Step-by-step explanation:
x : glass items
9x - 30 : plastic items
(the equation will be set equal to [tex]\frac{15}{85}[/tex] because 85% of the recyclables are plastic, so that means 15% must be glass.)
[tex]\frac{x}{9x-30} =\frac{15}{85}[/tex]
[tex]85x=135x-450[/tex] (now subtract 135x from each side to isolate x to one side of the equation)
[tex]-50x=-450[/tex] (divide each side of the equation by -50 to isolate x)
[tex]x=9[/tex]
value of (–3/8) × (+8/15).
A. 11/15
B. 11/23
C. –1/5
D. –1/3
ANSWER
The correct answer is C
EXPLANATION
We want to find the value of
[tex] - \frac{3}{8} \times \frac{8}{15} [/tex]
This is the same as:
[tex] - \frac{3}{8} \times \frac{8}{3 \times 5} [/tex]
We cancel out the common factors to get:
[tex]- \frac{1}{1} \times \frac{1}{1 \times 5} [/tex]
We simplify to get:
[tex] - \frac{1}{5} [/tex]
The correct answer is C
Determine whether the relation shown is a function. Explain how you know.
Answer:
It is not a function
Step-by-step explanation:
The plot shows (1, 1) and (1, 3) are both defined by the relation. It does not pass the "vertical line test", which requires the relation be single-valued everywhere.
Answer:
This is not a function
Step-by-step explanation:
To determine whether a relation is a function, we can use the vertical line test. If a vertical line touches the relation in more than one point, it is not a function.
Since a vertical line will touch the relation at two points at x=1
(1,1) and (1,3) this is not a function
Find the sum of the infinite geometric series, if it exists.
4 - 1 + 1/4 - 1/16 + . . .
a. -1
b. 16/5
c. 3
d. does not exist
So the series looks something like this:
[tex]\Sigma_{n=4}^{\infty}\frac{n-1}{16} \\
\frac{1}{16}\Sigma_{n=4}^{\infty}n-1[/tex]
If [tex]\lim_{n\rightarrow\infty}\neq[/tex] than [tex]\Sigma{a_n}[/tex] diverges. So we must apply limit infinity property:
[tex]\lim_{n\rightarrow\infty}(ax^n+\dots+bx+c)=\infty, a>0[/tex] and n is odd.
So...
[tex]\lim_{n\rightarrow\infty}n-1=\infty[/tex]
The series diverges.
Hope this helps.
r3t40
In a hospital ward, there are 15 nurses and 3 doctors. 9 of the nurses and 1 of the doctors are female. If a person is randomly selected from this group, what is the probability that the person is male or a doctor?
Answer:
i believe the answer is 9/18
draw a modelto show that x + 3 is the same as x over 3
Answer:
Step-by-step explanation:
Your equation comes out to x + 3 = x/3. Multiplying all three terms by 3 results in 3x + 9 = x, which simplifies to 2x = -9, so that x = -9/2.
Unsure of what you're asking for by "model."
greatest common factor.
35+50
Answer:
GCF is 5
Step-by-step explanation:
The factors of 35 are: 1, 5, 7, 35
The factors of 50 are: 1, 2, 5, 10, 25, 50
A.
10,000
B.
23,000
C.
55,000
D.
155,000
The best and most correct answer among the choices provided by the question is B.) 23,000
Hope this helps :)
A spinner with 6 equally sized slices is shown below. The dial is spun and stop on a slice at random . What is the probability that the dial stop at a grey side . Write your answer as a fraction in simplest form
One grey side or one gray side?
P(one grey side) = 1/6
The wheels on Jason's dirt bike measure 19 inches in diameter. How many revolutions will the wheels make when Jason rides for 500 feet? Use 3.14 for ?. Round to the nearest whole revolution.
Answer: 101 revolutions
Step-by-step explanation:
Answer: The total number of revolutions is 101.
Step-by-step explanation: Given that the wheels on Jason's dirt bike measure 19 inches in diameter.
We are to find the number of revolutions that the wheels make when Jason rides for 500 feet.
We know
1 foot = 12 inches.
So, 500 feet = 12 × 500 = 6000 inches.
Now, diameter of the wheels is given by
d = 19 inches.
So, the circumference of the wheels will be
[tex]C=\pi d=3.14\times19=59.66~\textup{inches}.[/tex]
Therefore, to ride 500 feet, The number of revolutions that the wheels make is given by
[tex]n=\dfrac{6000}{59.66}=100.56989...[/tex]
Rounding to the nearest whole number, the required number of revolutions will be 101.
Thus, the total number of revolutions is 101.
Which of the following represents another way to write the function rule f (x) = 3x + 1?
A. x = 3y + 1
B. f ( 1 ) = 4
C. f (y) = 3x + 1
D. y = 3x + 1
Answer:
d
Step-by-step explanation:
y=mx+b
Answer:
D. y = 3x+1
Step-by-step explanation:
A -- this describes the inverse function rule (x and y are interchanged).
B -- this describes one point that is described by the function rule, but does not cover the remaining infinite number of points.
C -- there is no variable y in the right side of the rule, so this would give f(1) = 3x+1, f(2) = 3x+1, and so on. The value will always be "3x+1" for any function argument. This is not the same as the given function rule.
D -- y (or any other variable except x) can stand in for f(x), so this is the same as y = f(x). It is another way to write the function rule.
Will give brainliest!
Aubrey was offered a job that paid a salary of $45,000 in its first year. The salary was set to increase by 2% per year every year. If Aubrey worked at the job for 20 years, what was the total amount of money earned over the 20 years, to the nearest whole number?
Aubrey's total earnings over 20 years, given a starting salary of $45,000 and a 2% annual increase, would be approximately $1,157,357, calculated using the formula for an increasing annuity.
Explanation:The subject of this question is compound interest, which is a mathematical concept used to calculate the future value of an investment or loan. In this case, Aubrey's salary is subject to an annual increase of 2%, which is compounded yearly.
To calculate the total amount earned over 20 years, we need to use the formula for the future value of a salary subject to yearly increases: Future Value = Salary * ((1 + Rate)^Years). Substitute the given values into the equation: Future Value = $45,000 * ((1 + 0.02)^20). Calculating this gives a value of approximately $66,207.11, which would be Aubrey's salary in the 20th year.
To find the total earnings over 20 years, you would add each year's salary together. However, it is much more complex due to the increasing salary. In this case, an annuity formula can be used.
The formula is: Total Earnings = Pmt * ((1 - (1 + r) ^ -n) / r). 'Pmt' is the first period payment, 'r' is the interest rate, and 'n' is the number of payments. Here, Pmt = $45,000, r = 2% or 0.02, and n = 20 years. After substituting everything in, the result should be around $1,157,357 rounded to the nearest whole number.
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URGENT WILL GIVE BRAINLIEST
Determine two pairs of polar coordinates for the point (5, 5) with 0° ≤ θ < 360°.
Answer:
(5sqrt(2), 45 deg)
(-5sqrt(2), 225 deg)
Step-by-step explanation:
(x,y)=(5,5)
So theta=arctan(5/5)=arctan(1)=45 degrees
Now r! r=sqrt(x^2+y^2)=sqrt(5^2+5^2)=sqrt(50)=sqrt(25)sqrt(2)=5sqrt(2)
So one polar point is (5sqrt(2) , 45 degrees)
Now if we do 180+45=225 degrees... this puts us in the 3rd quadrant... to get back to quadrant 1 we just take the opposite of our r
so another point is (-5sqrt(2) , 225 degrees)
What is the equation of the parabola with focus (1,1/2) and directrix y=3
Answers:
A. Y=-1/5x^2+2/5x+31/20
B. Y=-3/16x^2
C. Y=-3/5x^2-14/31x+71/16
D. Y=-x^2+17x-1
At which x-values are the output values of the floor function g(x) = ?x? and the ceiling function h(x) = ?x? equal? Check all that apply. –8 –5.2 –1.7 0 2.4
Answer:
-8, 0
Step-by-step explanation:
The floor and ceiling functions have equal values for any integer argument. Their value is that integer.
Among your answer choices, the integers are -8 and 0.
Answer:
-8 and 0 or A and D
Step-by-step explanation:
Just took it on Edge 2020
In order to unload clay easily, the body of a dump truck must be elevated to at least 50°. The body of a dump truck that is 15 feet long has been raised to 9 feet. Will the clay pour out easily? Show your work and draw a diagram to support your answer.
Please include the following:
• A diagram
• The trig equation you are solving and the steps you take to solve it.
• An answer in the context of the problem.
_________ (Yes or No), The clay ___________ (will or will not) pour out easily when the body of the dump truck is raised to 9 feet. I know this because ______________________________________.
Answer:
No; the angle of elevation of the front bottom of the dump truck body will be only 31°, much smaller an angle than the required 50°
Step-by-step explanation:
Let the body of the truck rest upon the x-axis with the rear body at (0, 0). The front of that body is located at (15, 0). The dump body is elevated to 9 ft.
We thus have a right triangle with base 15 ft and height 9 ft, and want to know what the angle is. The angle is at (0, 0). The front of the body is at (0, 15), and when elevated that point will be located at (15, 9).
The tangent function relates the angle to the opposite side (length 9 ft) and the adjacent side (length 15 ft):
tan Ф = opp / hyp = 9 / 15 = 31 degrees.
NO: the elevation of the front of the dump truck body is only 31° approx. The clay will not easily dump from the body.
Sameena measures the angle from the ground to the top of a building from two locations. The angle of elevation from the first point is 45°, and the angle of elevation from the second point is 60°. The distance from the second point to the top of the building is 150 ft.
[Not drawn to scale]
What is the distance from the first observation point to the top of the tower (x)? Round the answer to the nearest tenth.
122.5 ft
183.7 ft
244.9 ft
501.9 ft
Answer:
183.7 ft
Step-by-step explanation:
The mnemonic SOH CAH TOA reminds you that the relation between the hypotenuse and opposite side in a right triangle is ...
Sin = Opposite/Hypotenuse
For the first observation, the angle is 45° and the length of the hypotenuse of the triangle is given as x. For the second observation, the opposite side (building height) is the same, the angle is 60°, and the hypotenuse is 150 ft.
We can multiply the above equation by "Hypotenuse" to get an expression for "Opposite".
Opposite = Sin×Hypotenuse
For our two observations, this becomes ...
sin(45°)·x = (building height) = sin(60°)·(150 ft)
Dividing by the coefficient of x, we have ...
x = (150 ft)·sin(60°)/sin(45°) ≈ 183.7 ft
Answer:
B: 183.7
Step-by-step explanation:
First we need to find our values
A = 45°
B = 180 - 60 = 120
a = 150
Then we can solve for x using this formula x = a * sin(B)/sin(A).
x = 150 * sin(120)/sin(45)
x = 150 * √6/2
x = 75√6
x = 183.71173
The ideal temperature of a freezer to store a particular brand of ice cream is 0°F, with a fluctuation of no more than 2°F. Which inequality represents this situation if t is the temperature of the freezer?
Answer:
| x - 0 | ≤ 2
Step-by-step explanation:
Given,
The ideal temperature of the freezer = 0° F,
Also, it can fluctuate by 2° F,
Thus, if x represents the temperature of the freezer,
Then, there can be two cases,
Case 1 : x > 0,
⇒ x - 0 ≤ 2
Case 2 : If x < 0,
⇒ 0 - x ≤ 2,
⇒ -( x - 0 ) ≤ 2,
By combining the inequalities,
We get,
| x - 0 | ≤ 2,
Which is the required inequality.
This is a rational expression because the denominator contains a variable. This is a polynomial with 3 terms. This is a rational expression because the denominator contains a variable. This is a polynomial with 4 terms. This is a rational expression because the denominator contains a variable. This is a polynomial with 4 terms. This is a rational expression because the denominator contains a variable. This is a polynomial with 3 terms. This is a rational expression because the denominator contains a variable. This is a polynomial with 5 terms.
Answer:
i agree
Step-by-step explanation:
Given the functions, f(x) = 2x^2 + 2 and g(x) = 3x + 1, perform the indicated operation. When applicable, state the domain restriction.
A)2x2 - 3x + 1
B)2x2 - 3x + 3
C)5x3 + 2x2 + 2
D)6x3 + 6x + 3
Answer:
the one that equals to 7
Step-by-step explanation:
Answer: The answer is B 2x2 - 3x+3
Step-by-step explanation:
A jumbo crayon is composed of a cylinder with a conical tip. The cylinder is 12 cm tall with a radius of 1.5 cm, and the cone has a slant height of 2 cm and a radius of 1 cm. The lateral area of the cone is π cm2. To wrap paper around the entire lateral surface of the cylinder, π cm2 of paper is needed. The surface area, including the bottom base of the crayon, is π cm2.
Answer:
Part 1) The lateral area of the cone is [tex]LA=2\pi\ cm^{2}[/tex]
Part 2) The lateral surface area of the cylinder is [tex]LA=36\pi\ cm^{2}[/tex]
Part 3) The surface area of the crayon is [tex]SA=41.50\pi\ cm^{2}[/tex]
Step-by-step explanation:
Part 1) Find the lateral area of the cone
The lateral area of the cone is equal to
[tex]LA=\pi rl[/tex]
we have
[tex]r=1\ cm[/tex]
[tex]l=2\ cm[/tex]
substitute
[tex]LA=\pi (1)(2)[/tex]
[tex]LA=2\pi\ cm^{2}[/tex]
Part 2) Find the lateral surface area of the cylinder
The lateral area of the cylinder is equal to
[tex]LA=2\pi rh[/tex]
we have
[tex]r=1.5\ cm[/tex]
[tex]h=12\ cm[/tex]
substitute
[tex]LA=2\pi (1.5)(12)[/tex]
[tex]LA=36\pi\ cm^{2}[/tex]
Part 3) Find the surface area of the crayon
The surface area of the crayon is equal to the lateral area of the cone, plus the lateral area of the cylinder, plus the top area of the cylinder plus the bottom base of the crayon
Find the area of the bottom base of the crayon
[tex]A=\pi[r2^{2}-r1^{2}][/tex]
where
r2 is the radius of the cylinder
r1 is the radius of the cone
substitute
[tex]A=\pi[1.5^{2}-1^{2}][/tex]
[tex]A=1.25\pi\ cm^{2}[/tex]
Find the area of the top base of the cylinder
[tex]A=\pi(1.5)^{2}=2.25\pi\ cm^{2}[/tex]
Find the surface area
[tex]SA=2\pi+36\pi+2.25\pi+1.25\pi=41.50\pi\ cm^{2}[/tex]
Answer:
2
36
41.5
Step-by-step explanation:
just got it all right even though I guessed soooo, yay
Step-by-step explanation: