16. Write the product as a sum: 7 cos(6x)cos (7x) 17. Write the sum as a product: sin (2x) -sin (7x)

In response to the given query, the result we have is Hence, each expressions container contains 1/2 gallon of paint.

Multiplying, dividing, adding, and subtracting are all mathematical operations. As an example, consider the following expression: Expression, mathematics, and a numeric value Numbers, parameters, and functions are the components of an expression in mathematics. Using opposing words and phrases is possible. An expression, sometimes referred to as an algebraic expression, is any mathematical statement that includes variables, numbers, and a mathematical action between them. For instance, the expression 4m + 5 is made up of the expressions 4m and 5, as well as the variable m from the previous equation, all of which are separated by the mathematical symbol +.

The remaining paint would be as follows if the art teacher had 8 gallons and the students had used 3/4 of it:

8 x 1 - 3/4 = 8 x 1/4 = 2 gallons.

The 4 bottles of the remaining 2 gallons of paint each contain the following amounts of paint:

1/2 gallon is equal to 2/4.

Hence, each container contains 1/2 gallon of paint.

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Same-side interior angles are a pair of angles formed when two parallel lines are intersected by a third line, known as a transversal.

In this case, the same-side interior angles are the angles that are on the same side of the transversal and between the two parallel lines. They are called "interior" because they are inside the two parallel lines.

If you take any two same-side interior angles, they will add up to 180 degrees, meaning that they are supplementary.

The same side interior angles for this problem are given as follows:

As the same side interior angles are supplementary, the value of x is obtained as follows:

4x - 23 + 2x + 5 = 180.

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The function g(x) is given as follows:

To shift the exponential function f(x) = -7x+1 two units to the right, we need to replace x with (x-2) in the equation.

This is because when x is replaced with (x-2), the function will have the same output value as before when x was 2 units less.

Thus, the shifted function g(x) can be written as:

g(x) = f(x-2)

g(x) = -7(x-2) + 1

g(x) = -7x + 15

So, the function g(x) is given by g(x) = -7x + 15, and it is the same as f(x) shifted 2 units to the right.

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Answer:

Ava has 25 books initially, then gives away 7 books, leaving her with 25 - 7 = 18 books.

After Tom gives her 12 books, she will have 18 + 12 = 30 books.

Therefore, Ava has 30 books now.

Answer:

She would have 30 books

Step-by-step explanation:

25 - 7 + 12 = 30

The missing value that cοmpletes the frequency table is 174.

A frequency distributiοn table is a table that displays the number (οr frequency) οf οbservatiοns that fall within each interval after grοuping the data intο intervals. It is an effective tοοl fοr cοmprehending data distributiοn and spοtting patterns οr trends.

Typically, the table has twο cοlumns. The intervals, οr "bins," intο which the data is divided are listed in the first cοlumn. The frequency, οr quantity οf οbservatiοns, that fall intο each bin is displayed in the secοnd cοlumn. Depending οn the data and the level οf precisiοn required, the bins may be identical οr vary in width.

The analysis οf a variety οf data types, including numerical, categοry, and οrdinal data, can be dοne using frequency distributiοn tables. As well as calculating descriptive statistics like the mean, median, and mοde, as well as measures οf variability like the range and standard deviatiοn, they can alsο be used tο generate these statistics.

All things cοnsidered, frequency distributiοn tables οffer a quick and effective apprοach tο summarize a lοt οf data and learn mοre abοut its distributiοn and prοperties.

Tο cοmplete the frequency table, we need tο determine the frequency fοr the interval 28 ≤ x < 30.

We can see frοm the histοgram that the interval 28 ≤ x < 30 has a width οf 2. Since the tοtal frequency is the sum οf the frequencies in each interval, we can use the infοrmatiοn prοvided in the frequency table tο sοlve fοr the missing value:

110 + 36 + 180 + frequency fοr 28 ≤ x < 30 = tοtal frequency

We can simplify this expressiοn:

326 + frequency fοr 28 ≤ x < 30 = tοtal frequency

We knοw that the tοtal frequency is the number οf mοbile phοnes that were tested, sο we can assume that this is a knοwn quantity. Let's say that the manufacturer tested 500 mοbile phοnes. Then we can sοlve fοr the missing frequency:

326 + frequency fοr 28 ≤ x < 30 = 500

frequency fοr 28 ≤ x < 30 = 174

Therefοre, the missing value that cοmpletes the frequency table is 174. The cοmpleted frequency table is:

Battery life, x(hοurs) Frequency

15 < x < 20                   110

20 ≤ x < 22                  36

22 ≤ x < 28                  180

28 ≤ x < 30                   174

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The answer of the given question based on the  image of the given rotation of the preimage is given below,

Rotation is a transformation in which an object or figure is turned around a fixed point or center by a certain angle or degree. The fixed point is known as the center of rotation, and the angle of rotation is measured in degrees or radians.

Based on the given rotation specification, "r(270,0)(x,y)", the preimage should be rotated 270 degrees counterclockwise around the origin.

To draw the image of the rotated preimage, you can use the following steps:

Plot the coordinates of the preimage points on a coordinate plane.

Draw  line from of each point to origin.

Measure an angle of 270 degrees counterclockwise from each line, using a protractor or angle tool.

Draw a new line from each point to the endpoint of the measured angle. These lines represent the rotated points.

Label the new coordinates of the rotated points.

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The inequality is represented as t + b ≤ 102 , where t is the number of tacos and b is the number of burritos

Inequalities are the mathematical expressions in which both sides are not equal. In inequality, unlike in equations, we compare two values. The equal sign in between is replaced by less than (or less than or equal to), greater than (or greater than or equal to), or not equal to sign.

In an inequality, the two expressions are not necessarily equal which is indicated by the symbols: >, <, ≤ or ≥.

Given data ,

Let the inequality equation be represented as A

Now , the value of A is

Substituting the values in the equation , we get

Let the number of tacos be = t

Let the number of burritos be = b

The total number of burritos or tacos be = 102

On simplifying , we get

t + b = 102

So , the maximum number of tacos and burritos to be sold is

t + b ≤ 102

Hence , the inequality is t + b ≤ 102

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For both march and April the precipitation is [tex](4\frac{7}{16})[/tex] inches below the average.

The term "average" refers to a central value that represents a set of numbers. There are different types of averages, such as the mean, mode and median.

Let's denote the average precipitation as "a".

In March, the precipitation was [tex]6\frac{1}{2}[/tex] inches less than the average. This can be expressed as:

March precipitation = [tex]a-6\frac{1}{2}[/tex]

In April, the precipitation was [tex]2\frac{3}{8}[/tex] inches less than the average. This can be expressed as:

April precipitation = [tex]a-2\frac{3}{8}[/tex]

To find the total precipitation below average for both March and April, we can add the two expressions:

March precipitation + April precipitation = [tex](a-6\frac{1}{2})[/tex] + [tex](a-2\frac{3}{8})[/tex]

Simplifying the expression, we get:

March precipitation + April precipitation = [tex](2a-8\frac{7}{8})[/tex]

So, after dividing by 2 in result the value will be, [tex](a-4\frac{7}{16})[/tex]

Therefore, the total precipitation below average for both March and April was [tex](4\frac{7}{16})[/tex] inches.

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The number that if decreased by one fifth of itself gives 132 is 165.

To solve this problem, we can use algebra to create an equation and then solve for the unknown number.

Let x be the number we are trying to find.

If the number decreased by one-fifth of itself yields 132, we can write the equation:

x - (1/5)x = 132

Simplifying the equation, we get:

(4/5)x = 132

Multiplying both sides by 5/4 to isolate x, we get:

x = (5/4)(132)

x = 165

Therefore, the number that decreased by one-fifth of itself yields 132 is 165.

Alternatively, we can use a different method to solve the problem.

If the number decreased by one-fifth of itself yields 132, we can write the equation:

x - (x/5) = 132

Multiplying both sides by 5 to eliminate the fraction, we get:

5x - x = 660

Simplifying the equation, we get:

4x = 660

Dividing both sides by 4 to isolate x, we get:

x = 165

Therefore, the number that decreased by one-fifth of itself yields 132 is 165.

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The volume of the composite figure is 301 cm³

Volume is defined as the space occupied within the boundaries of an object in three-dimensional space. It is also known as the capacity of the object.

Given is shape of ice creme we need to find the shapes in it and the volume.

1) The figure is a composite figure of a hemisphere and a cone.

2) Volume of hemisphere = 2/3 π × radius³, here radius is 4 cm

Volume = 2/3 π × 4³

= 2/3 × 3.14 × 64

= 134 cm³

Volume of a cone  = 1/3 π × radius² × height, here height is 10 cm

Volume = 1/3 × 3.14 × 16 × 10

= 167 cm³

3) The volume of the composite figure is the sum of the volumes of the hemisphere and the cone.

= 134 + 167 = 301 cm³

Hence, the volume of the composite figure is 301 cm³

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Answer:

Step-by-step explanation:

Given: [tex]f(x) = 7x^{2} -10x+10[/tex]

To find: Range of the function

To find the range of the function, write the quadratic equation in the form of vertex form of parabola, that is:

[tex]f(x) = a(x-h)^{2} +k[/tex]

If a > 0, range of the function of [k, ∞)

[tex]f(x) = 7x^{2} -10x+10=7(x^{2} -\frac{10}{7} x+\frac{10}{7} )[/tex]

[tex]f(x) = 7(x-\frac{5}{7} )^{2} +\frac{45}{7}[/tex]

by comparing this expression with the vertex form of parabola, we can say that the range of the function is [[tex]\frac{45}{7}[/tex], ∞)

The range of the quadratic function f(x) = 7x2 – 10x + 10 is {y | y ≥ 315/49}.

The range of a quadratic function is the set of all possible values that the function can take on. To find the range of the given quadratic function f(x) = 7x2 – 10x + 10, we need to find the vertex of the function, which is the point at which the function reaches its maximum or minimum value.

The x-coordinate of the vertex of a quadratic function is given by:
x = -b/(2a)

Where a and b are the coefficients of the quadratic term and the linear term, respectively. In this case, a = 7 and b = -10, so:
x = -(-10)/(2*7) = 10/14 = 5/7

Now, we can plug this value of x back into the function to find the y-coordinate of the vertex:
y = 7(5/7)2 – 10(5/7) + 10 = 175/49 – 50/7 + 10 = 175/49 – 350/49 + 490/49 = 315/49

So the vertex of the function is at the point (5/7, 315/49).

Since the coefficient of the quadratic term is positive, the function opens upwards, and the vertex is the minimum point of the function. Therefore, the range of the function is:
y ≥ 315/49

In inequality form, the range of the quadratic function f(x) = 7x2 – 10x + 10 is {y | y ≥ 315/49}.

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14) all triangles add up to 180 degrees.

for example: c - 40 + 40 = 80; 180-80 = 100

16)

a. EF

b. E

c. A

We have the following response after answering the given question: function Hence, the quarterly growth rate is almost 4.08%.

Mathematicians research numbers, their variants, equations, forms, and related structures, as well as possible locations for these things. The relationship between a group of inputs, each of which has a corresponding output, is referred to as a function. Every input contributes to a single, distinct output in a connection between inputs and outputs known as a function. A domain, codomain, or scope is assigned to each function. Often, functions are denoted with the letter f. (x). The key is an x. There are four main categories of accessible functions: on functions, one-to-one capabilities, so many capabilities, in capabilities, and on functions.

[tex]P(t) = P0 * e^(rt) (rt)[/tex]

Where q is the quarterly rate of increase, [tex]q = (1 + r)(1/4) - 1.[/tex]

[tex]q = (1 + 0.17)^(1/4) - 1 ≈ 0.0408\sP(t) = 1100 * e^(0.17t) (0.17t)[/tex]

Q is equal to round[tex]((1 + 0.17 ** (1/4) - 1, 4)[/tex]

P0 = 1100

math.log(1 + q) = r

(q * 100, 2)

4.08 P(t) = P0 * e(rt), where r is the natural logarithm's base and P0 is the beginning population (approximately 2.71828).

Where q is the quarterly rate of increase, q = (1 + r)(1/4) - 1.

[tex]q = (1 + 0.17)^(1/4) - 1 ≈ 0.0408\sP(t) = 1100 * e^(0.17t) (0.17t)[/tex]

We may convert the quarterly growth rate to a percentage and round to the closest hundredth of a percent to get the percentage rate of change every quarter:

(q * 100, 2)

4.08

Hence, the quarterly growth rate is almost 4.08%.

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Answer:

Step-by-step explanation:

2b + 3d = 13.25

2b + 2d = 11.50

2b + 3d = 13.25

-2b - 2d = -11.50

d = $1.75 for one drink

2b + 3.50 = 11.50

2b = 8

b = $4 for one Big Mac

Answer:

Step-by-step explanation:

[tex]\frac{-1}{12}[/tex]

Answer:

a

Step-by-step explanation:

- [tex]\frac{1}{4}[/tex] × [tex]\frac{2}{3}[/tex] × [tex]\frac{1}{2}[/tex] ( cancel the 2 on the numerator/ denominator of last 2 fractions )

= - [tex]\frac{1}{4}[/tex] × [tex]\frac{1}{3}[/tex] × 1

multiply values on numerators and values on denominators )

= - [tex]\frac{1(1)}{4(3)}[/tex] × 1

= - [tex]\frac{1}{12}[/tex]

Answer:

We can use a recursive approach to solve this problem. Let's define a function $f(n)$ as the number of ways to ascend a flight of $n$ steps, where the priestess can only take one or two steps at a time.

To ascend a flight of 1 step, there is only one way to do it (take one step). Therefore, $f(1) = 1$.

To ascend a flight of 2 steps, there are two ways to do it (take two steps at once or take one step twice). Therefore, $f(2) = 2$.

For a flight of $n$ steps, the priestess can either take one step and ascend the remaining $n-1$ steps in $f(n-1)$ ways, or take two steps and ascend the remaining $n-2$ steps in $f(n-2)$ ways. Therefore, we have the recursive formula:

$$f(n) = f(n-1) + f(n-2)$$

Using this formula, we can calculate the number of ways to ascend a flight of 5 steps:

$$f(1) = 1$$

$$f(2) = 2$$

$$f(3) = f(2) + f(1) = 3$$

$$f(4) = f(3) + f(2) = 5$$

$$f(5) = f(4) + f(3) = 8$$

Therefore, there are 8 different ways for the priestess to ascend the steps. To find out if she can do this a different way each day for a week, we need to make sure that there are at least 7 different ways. Since there are 8 ways, it is possible for the priestess to ascend the steps in a different way each day for a week.

Answer:

Height = 8

Volume = 48

Step-by-step explanation:

(4x4)x3 = 48

The distance between the points (-9, 4) and (3, 12) is approximately 14.42 units.

what exactly is distance?

Distance is the amount of space between two objects or locations. It is a measure of how far apart two things are from each other, typically expressed in units such as meters, kilometers, miles, feet, or yards. Distance can be measured in a straight line, known as the "Euclidean distance," or it can be measured along a path or route, known as the "distance traveled." Distance is a fundamental concept in many areas of science and mathematics, including physics, geography, and geometry

The formula for calculating the distance between two points is d = ((x2 - x1)2 + (y2 - y1)2).

Let's use this formula to find the distance between the two points (-9, 4) and (3, 12):

d = √((3 - (-9))² + (12 - 4)²)

= √(12² + 8²)

= √(144 + 64)

= √208

≈ 14.42

So the distance between the points (-9, 4) and (3, 12) is approximately 14.42 units.

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The function f(x) = 3x² + 24x + 48 has a zero with a multiplicity of 2.

To determine which function has a zero with a multiplicity of 2, we need to find the discriminant of each quadratic function and check if it is equal to zero.

The discriminant is given by the formula: b² - 4ac, where a, b, and c are the coefficients of the quadratic equation ax² + bx + c = 0.

Let's calculate the discriminant for each function:

1) f(x) = -3x² + 12x + 12

Discriminant: b² - 4ac = (12)² - 4(-3)(12) = 144 + 144 = 288 (not zero)

2) f(x) = 3x² + 24x + 48

Discriminant: b² - 4ac = (24)² - 4(3)(48) = 576 - 576 = 0 (zero)

3) f(x) = x² + 3x + 9

Discriminant: b² - 4ac = (3)² - 4(1)(9) = 9 - 36 = -27 (not zero)

4) f(x) = x² - 2x - 1

Discriminant: b² - 4ac = (-2)² - 4(1)(-1) = 4 + 4 = 8 (not zero)

Based on the calculations, the function f(x) = 3x² + 24x + 48 has a zero with a multiplicity of 2.

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There is a 0.0876 percent probability that six or more pages will include mistakes. There is a 1.64 x 10⁻⁶ chance that more than 10 pages may include mistakes.

Information about possibility of occurrence is provided by probability. For instance, weather patterns are used by meteorologists to predict the possibility of rain. In order to understand the relationship between exposures and the risk of health outcomes, epidemiology uses probability theory.

a) Chance that six or more pages may have mistakes:

To get this probability, we can utilise the binomial probability formula:

P(X>=6) = 1-P(X<6)

where X is the number of pages with errors.

P(X<6) = P(X=0)+P(X=1)+P(X=2) +P(X=3)+P(X=4)+P(X=5)

For each term, using the binomial probability formula, we obtain:

P(X < 6) = 0.9124

Therefore, P(X >= 6) = 1 - P(X < 6) = 1 - 0.9124 = 0.0876

Hence, there is a 0.0876 percent chance that six or more pages will include errors.

b) Chance that there are errors on more than 10 pages:

Using the same method, we can discover:

P(X > 10) = P(X = 11) + P(X = 12) + ... + P(X = 200)

To find the likelihood, we can use a calculator or software because performing the math by hand can be very time-consuming. Using a calculator for the binomial distribution, we obtain:

P(X > 10) = 1.64 x 10⁻⁶ (rounded to 3 decimal places)

Hence, there is a roughly 10% chance that there will be errors on more than 10 pages. 1.64 x 10⁻⁶.

c) Probability that none of the pages contain errors:

P(X = 0) can be calculated using the binomial probability formula:

P(X = 0) = (200 choose 0) * (0.009)⁰ * (1 - 0.009)²⁰⁰

= 0.8196

As a result, 0.8196 percent of the pages are likely to be error-free.

d) Probability that fewer than five pages contain errors:

P(X<5)=P(X=0)+P(X=1)+P(X=2)+P(X=3)+P(X=4)

Again, using the binomial probability formula for each term, we get:

P(X < 5) = 0.8151

Hence, the likelihood that fewer than five pages are mistake-free is 0.8151.

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The correct option is (c) $768x^{\frac{1}{3}}$

The given expression is as follows; \[ \left(\frac{3 \sqrt[3]{x y} y^{-3}}{\sqrt[3]{y} x^{-\frac{4}{3}}}\right)^{-1} \]Firstly, simplify the numerator and the denominator of the expression.\[\frac{3\sqrt[3]{xy}y^{-3}}{\sqrt[3]{y}x^{-\frac{4}{3}}}=\frac{3}{\sqrt[3]{x}x^{-\frac{4}{3}}\sqrt[3]{y}y^{-3}}\]Next, simplify the denominator of the expression.\[\frac{3}{\sqrt[3]{x}x^{-\frac{4}{3}}\sqrt[3]{y}y^{-3}}=3x^{\frac{1}{3}}y^{\frac{10}{3}}\]Now, let us substitute $y=16$ into the expression of $3x^{\frac{1}{3}}y^{\frac{10}{3}}$.\[3x^{\frac{1}{3}}y^{\frac{10}{3}}=3x^{\frac{1}{3}}(16)^{\frac{10}{3}}=48x^{\frac{1}{3}}(2)^{\frac{10}{3}}=768x^{\frac{1}{3}}\]As we obtained the value of the expression by substituting $y=16$, therefore the correct option is (c) $768x^{\frac{1}{3}}$.

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Ian’s mom sang him S songs. Then his sister sang him two times as many as his mom had. If Ian listened to nine songs in all, Ian's mom sanghim twIan’s mom sang him S songs. Then his sister sang him two times as many as his mom had. If Ian listened to nine songs in all, Ian's mom sango times as many as his mom had. If Ian listened to nine songs in all, Ian's mom sang _____songs.

Answer: (x - 4, y - 5)

Step-by-step explanation: (x, y) --> (x - 4, y - 5)

The final velocity of the jet flying in the north direction after accelerating for 15s is 473 m/s.

When observed from a specific point of view and as measured by a specific unit of time, velocity is the direction at which an item is moving and serves as a measure of the pace at which its position is changing. How quickly or slowly an object is travelling can be determined by its velocity and speed. Being a vector quantity, we need to define velocity in terms of both magnitude (speed) and direction. A body is considered to be accelerating if the magnitude or direction of its velocity changes.

Given,

The initial velocity u = 200 m/s

Acceleration of jet a = 18.2 m/s²

Time taken t = 15s

We are asked to find the final velocity v of the jet.

W can use the following formula to find the final velocity.

v = u+ at

  = 200 + (18.2) × 15

 = 473 m/s (north)

Therefore the final velocity of the jet flying in the north direction after accelerating for 15s is 473 m/s.

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The plane will end up flying 1.7 degrees off course.
The plane's speed relative to the ground will be 729.1 km/hr.

To determine the plane's speed and direction relative to the ground, we need to use vector addition to combine the velocity of the plane and the velocity of the wind.

First, let's find the components of the wind velocity. Since the wind is blowing from the southeast, its x-component is -30cos(45) = -21.2 km/hr and its y-component is -30sin(45) = -21.2 km/hr.

Next, let's find the components of the plane's velocity. Since the plane is heading north, its x-component is 0 km/hr and its y-component is 750 km/hr.

Now, we can add the components of the plane's velocity and the wind's velocity to find the plane's velocity relative to the ground:

x-component: 0 + (-21.2) = -21.2 km/hr

y-component: 750 + (-21.2) = 728.8 km/hr

Finally, we can use the Pythagorean theorem to find the magnitude of the plane's velocity relative to the ground:

speed = sqrt((-21.2)^2 + (728.8)^2) = 729.1 km/hr

And we can use the inverse tangent function to find the direction of the plane's velocity relative to the ground:

direction = tan^(-1)(-21.2/728.8) = -1.7 degrees

So, the plane will end up flying 1.7 degrees west of north, and its speed relative to the ground will be 729.1 km/hr.

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The domain and range of the function are respectively, (-6, 6) & (0, 6).

The domain of a function is the set of all input values (independent variable) for which the function is defined and produces a valid output (dependent variable).

The range of a function is the set of all possible output values of the function. It represents the set of all possible values of the dependent variable.

Given that,

The graph of the function,

As we know from the definition of the graph,

the domain is all the possible values of the function for which it gives definite value so it can be seen in the graph,

function gives definite value only in the interval (-6, 6)

The range is all the outputs for the input value of  domain

and it can be seen in the graph,

the output values are in the range of (0, 6)

Therefore, the domain and range are respectively (-6, 6) & (0,6)

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A store has three sizes of cans of nuts. The large size contains 2 pounds of peanuts and 1 pound of cashews. The mammoth size contains 1 pound of walnuts, 6 pounds of peanuts, and 2 pounds of cashews.

The giant size contains 1 pound of walnuts, 4 pounds of peanuts, and 2 pounds of enehews. Suppose that the store recelives be filled with the given sizes of cans?

To solve this problem, let's assign variables to the unknowns. Let G represent the number of giant cans, M represent the number of mammoth cans, and L represent the number of large cans.

We can then write out the system of linear equations:

2L + 1M + 4G = 2 lbs. of peanuts
1L + 6M + 1G = 1 lbs. of walnuts
1L + 2M + 2G = 2 lbs. of cashews

Solving this system of equations can be done with any method, such as substitution, elimination, or graphing.

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