The formula for the sum of all terms is [tex]S_n[/tex]= n(n+1)/2.
Sum most commonly means the whole of two or greater numbers delivered collectively; see addition. A summation additionally known as a sum is the result of arithmetically adding numbers or portions. A summation constantly includes a whole range of terms. There can be as few as two terms, or as many as one hundred, a thousand, or one million. some summations include infinitely many terms.
In arithmetic, the summation is the addition of a chain of any form of numbers, called addends or summands; the result is their sum or general. the combination of two or extra numbers, magnitudes, portions, or details as decided via or as if via the mathematical technique of addition: The sum of 6 and 8 is 14. a selected combination or overall, specifically with regards to money: The fees came to a giant sum.
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Find the coordinates of the circumcenter of AABC with vertices A (1, 0), B (8, -7), and
(7,-4).
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).
What is circumcentre?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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Four students were asked to convert the polar coordinates (4, 11pi/6) to a complex number using technology. Select the correct work.
The complex number z = (4, 11π / 6) in rectangular form is z = 2√3 - i 2 (z = 3.464 - i 2). (Correct choice: C)
How to convert a complex number into rectangular formHerein 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:
r - Normθ - Direction, in radians.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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Write a quadratic equation in the form x^2+bx+c=0 that has the following roots:
Roots: −6±5i
The asked quadratic equation is x²+6x+41/2 = 0
What are quadratic equations?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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Solve for x in this figure.
Enter your answer in the box.
Previous
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°.
32 windmill each complete 120 revolution every minute. Which operation would you ue to find the total number of revolution the 32 windmill complete in 1 minute
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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An incident light ray strikes water at an angle of 20 degrees. The index of refraction of air is 1. 0003, and the index of refraction of water is 1. 33.
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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The drawing below shows three squares joined at their vertices to form a right triangle. The area of square A is 9 ft2 and the area of
square B is 16 ft². What is the side length of square C?
25
7
8
5
The side-length of square C as required in the task content is; 5 ft.
What is the side length of square C?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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In the triangle below,
x = [ ? ] cm. Round to the
nearest tenth.
15 cm
35
90°
The value of side x for the right-angle triangle is 8.6 cm.
What is trigonometry?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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Please help !!
Consider the line y = -2-4x.
What is the slope of a line perpendicular to this line?
What is the slope of a line parallel to this line?
Slope of a perpendicular line:
Slope of a parallel line:
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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Which equation represents the graph?
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
The figure shows a line graph and two shaded triangles that are similar:
Which statement about the slope of the line is true?
A.) It is 1/6 throughout the line.
B.) It is 6 throughout the line.
C.) The slope from point O to point A is 1/6 times the slope of the line from point A to point B.
D.) The slope from point O to point A is six times the slope of the line from point A to point B.
Answer: A). It is 1/6 throughout the line
Step-by-step explanation:
Help me write a simplified expression for the perimeter of a square, please.
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.
Unit 4 Lesson 5 Cumulative Practice Problems
1. The table shows the monthly revenue of a business rising exponentially since it
opened an online store.
(f)0
(f)3
months since online monthly revenue
store opened
in dollars
0
1
3
4
6
72,000
90,000
112,500
a. Describe how the monthly revenue is growing.
b. Write an equation to represent the revenue, R, as a function of months, m, since
the online store opened.
c. Find the monthly revenue 1 month after the online store opened. Record the
value in the table. Explain your reasoning.
d. Explain how we can use the value of R(1) to find R(4).
The company's monthly revenue rises by 8% on a monthly basis.
What is an exponential function?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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work out the answer to these simultaneous equations
y=4x
y=x² - 12
A doctor's office is deciding between buying 2 different sized cups. The 2 cups are shown. 1.5 in 4 in. --2 in.--- 6 in. Before deciding which size cup to buy, they determine how much water each size cup can hold. How much more water, in cubic inches, can the larger cup hold than the smaller cup? Round your answer to the nearest hundredth. Use the pi key on your calculator for pi. (On the EOC, this problem specifically says - in cubic inches so you need to add that to your answer. EX. 5.81 cubic inches)
The water that the larger cup can hold more than the smaller cup, given the two cups is 15. 71 cubic inches
How to find the amount of water ?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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A cell phone company uses the equation C=$0.15t+$35.00 to determine the total cost, C, for a month of service based on the number of text messages, t. Identify the slope.
Slope is $0.15 of the equation of cost C=$0.15t+$35.00.
What is Slope of Line?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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I need help with this. Please don't rush and take your time.
Answer:
The correct answer is B.
Find (−7z−4)−(−3z+1) .
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.
What is the solution for Y =- 3x 8?.
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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How many solutions does Y =- 3x 7 have?.
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?
What are the 5 types of inequality?.
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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in a cookie mix, we find walnut cookies, raisin cookies, and butter cookies in a ratio of 1: 2: 2. If a bag of the mix contains 30 raisin cookies, how many cookies in total are there?
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.
How do you tell if a word problem is proportional?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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A bulk food store sells two types of trail mix that are a combination of nuts, chocolate, and dried fruit. One of the trail mixes is 75% nuts, and the other is 35% nuts. A customer puts in a special order for 10 pounds of trail mix that is 60% nuts. How much of each trail mix should be combined to create the special order trail mix?
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.
What is an equation?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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What is the distance between the pair of points 2 3 and 4 1?.
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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What is the quotient 3x3 10x2 10x 4 )/( x 2?.
After solving, the quotient (3x^3 + 10x^2 + 10x + 4) ÷ (x + 2) is 3x^2 + 4x + 2.
In the question, we have to find the quotient (3x^3 + 10x^2 + 10x + 4) ÷ (x + 2).
The polynomial is 3x^3 + 10x^2 + 10x + 4.
We have to divide the given equation by x + 2 to find the quotient.
Four operations, namely divide, multiply, subtract, and bring down, can be used to break down the division process into phases.
1: Find the appropriate integer to use as the divisor for the given number.
2: Multiply the integer (quotient) and divisor to obtain the amount to be deducted from the payout.
3: Deduct the amount from the dividend.
4: Bring down the balance and any additional digits from the dividend in step four.
5: Continue the procedure outlined above until the remainder equals zero or less than the divisor.
x+2 ) 3x^3 + 10x^2 + 10x + 4 ( 3x^2 + 4x + 2
3x^3 + 6x^2
- -
___________
4x^2 + 10x
4x^2 + 8x
- -
______________
2x + 4
2x + 4
- -
____________
0
Hence, the quotient (3x^3 + 10x^2 + 10x + 4) ÷ (x + 2) is 3x^2 + 4x + 2.
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The complete question is:
What is the quotient (3x^3 + 10x^2 + 10x + 4) ÷ (x + 2)?
What is the slope of 2y =- 3x 1?.
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 :
2y = -3x + 1y = [tex]\frac{-3}{2}[/tex] x + [tex]\frac{1}{2}[/tex]Comparing this equation to standard equation, we find m = -3/2.
Therefore, the slope of given line is -3/2.
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PLSSS HELP TURLY KNOW THISSS
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.
(PLEASE HELP QUICKLY) Aiden and his children went into a grocery store and he bought $14 worth of apples and bananas. Each apple costs $1.75 and each banana costs $0.70. He bought a total of 14 apples and bananas altogether. By following the steps below, determine the number of apples, x, and the number of bananas, y, that Aiden bought.
Determine three ways to have a total of 14 apples and bananas
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
2. A lemonade recipe calls for cup of lemon juice for every cup of water.
a. Use the table to answer these questions.
i. What does x represent?
ii. What does y represent?
iii. Is there a proportional relationship between x and y?
b. Plot the pairs in the table in a coordinate plane.
X
1
2
3
4
y
4
1
2
3
²14
1
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:
I need help please———————————————————
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
