The approach Caprice uses is valid because it relies on the fundamental property of equations: if you add or subtract the same quantity from both sides, the equality is preserved. This means that the relationships described by the equations remain unchanged, allowing you to eliminate one variable (in this case, the y-terms) and simplify the system.

When equations A and B are added together, the operation combines their expressions while maintaining the truth of the equations. This makes it easier to solve for the remaining variable (x, in this instance) and ultimately leads to the correct solution.

So in summary, the addition of equations is valid because it keeps the balance of the equations intact, enabling effective elimination of variables. Keep up the good work! If you need more in-depth explanations, feel free to check the extended services page!

To find the radius ( r ) of the smaller cylinder when the volume ( V ) is 160π cm³, we start with the given equation:

[

160π = 250π – 10πr²

]

Now, let’s simplify the equation step by step:

1. Subtract 250π from both sides:

[

160π – 250π = -10πr²

]

This simplifies to:

[

-90π = -10πr²

]

2. Divide both sides by -10π (noting that π is positive and can be canceled out):

[

9 = r²

]

3. Take the square root of both sides:

[

r = 3

]

So, the correct solution is:

– ( r = 3 )

This means the radius of the smaller cylinder is 3 centimeters when the volume outside the smaller cylinder but inside the larger cylinder is 160π cm³.

To find out how much mint you need for a half recipe of 2 2/3 teaspoons, you can follow these steps:

1. First, convert the mixed number to an improper fraction.

– 2 2/3 can be converted as follows:

– 2 * 3 + 2 = 6 + 2 = 8

– So, 2 2/3 = 8/3 teaspoons.

2. Now, to find half of that quantity, divide by 2:

– (8/3) ÷ 2 = (8/3) * (1/2) = 8/6.

3. Finally, simplify 8/6:

– 8/6 = 4/3 teaspoons.

So, for a half recipe, you would need 4/3 teaspoons of mint, which can also be expressed as 1 1/3 teaspoons.

The correct equation is:

84 = 180 – 16t²

t = √6

### Explanation:

This equation models the height of the object over time, where 180 is the initial height, and 16t² represents the effect of gravity on the height. We set the height equal to 84 feet to find the time (t). When you solve for t, you find that t = √6, which tells you how long it takes for the object to reach a height of 84 feet. Keep practicing, and check out our extended services page if you need more help!

To find out how much farther the beetle crawled than the spider, we subtract the distance the spider crawled from the distance the beetle crawled.

1. The distance the beetle crawled: 2 yards
2. The distance the spider crawled: 1/5 yard

Now we perform the subtraction:

[

text{Distance beetle crawled} – text{Distance spider crawled} = 2 – frac{1}{5}

]

To subtract these, we need to convert 2 yards into a fraction with a denominator of 5:

[

2 = frac{10}{5}

]

Now we can subtract:

[

frac{10}{5} – frac{1}{5} = frac{10 – 1}{5} = frac{9}{5}

]

So, the beetle crawled (frac{9}{5}) yards farther than the spider.

This can also be expressed as a mixed number:

(frac{9}{5} = 1 frac{4}{5})

Thus, the beetle crawled (1 frac{4}{5}) yards farther than the spider.

To find out how much farther away Planet X is compared to Planet Y, we need to subtract the distance of Planet Y from the distance of Planet X.

1. Distance of Planet X = 1/6 light-year
2. Distance of Planet Y = 1/12 light-year

To subtract these fractions, we need a common denominator. The least common multiple of 6 and 12 is 12.

Now, let’s convert both fractions:

– For Planet X:

(frac{1}{6} = frac{2}{12}) (by multiplying the numerator and denominator by 2)

– For Planet Y:

(frac{1}{12} = frac{1}{12})

Now we can perform the subtraction:

(frac{2}{12} – frac{1}{12} = frac{1}{12})

So, Planet X is (frac{1}{12}) of a light-year farther away from Planet Y.

To find out how many pounds of bird seed Alexa can put in each bird cage, you need to divide the total amount of bird seed by the number of cages.

1. Total bird seed: ( frac{3}{4} ) pound.
2. Number of cages: 2.

Now, divide ( frac{3}{4} ) by 2. Dividing by 2 is the same as multiplying by ( frac{1}{2} ).

So, the calculation is:

[

frac{3}{4} times frac{1}{2} = frac{3 times 1}{4 times 2} = frac{3}{8}

]

Thus, Alexa can put ( frac{3}{8} ) pounds of bird seed in each bird cage.

The most efficient first step to solve the set of equations is to combine like terms in the first equation.

Let’s simplify the first equation:

1. Start with the equation: (y + 3x + 4y = 2x + 4 + 17).
2. Combine like terms on the left side: (5y + 3x = 2x + 21).
3. Now you can isolate variables and proceed from there.

This approach helps clarify the relationship between the variables right at the beginning, making it easier to solve the equations thereafter. For further assistance, check the extended services page for additional help!

For the system of equations provided, we can match the true statement to each case:

1. Infinitely Many Solutions: Both equations are equal to y.

– Explanation: If both equations represent the same line in a linear system, they will have infinitely many solutions because they overlap completely.

2. Zero Solutions: 6x – 3 can never be equal to 6x + 3.

– Explanation: This is true because simplifying both sides leads to a contradiction (subtracting 6x results in -3 = 3, which is false), indicating no solutions exist.

3. One Solution: 2x + 1 = 6x + 3 has one solution.

– Explanation: This equation can be solved algebraically, and it results in a single value for x, representing one unique solution.

4. Zero Solutions: y is equal to two different expressions.

– Explanation: If y is expressed as two different values (e.g., y = f(x) and y = g(x), where f(x) ≠ g(x)), it indicates that there is no point where both functions intersect, leading to zero solutions.

5. Infinitely Many Solutions: 6x + 3 is a multiple of 2x + 1.

– Explanation: This implies that one equation can be derived from the other through multiplication, suggesting infinitely many solutions as the lines coincide.

Overall, the correct matches

One Solution: The graphs of these lines would intersect at a point.

This means the two equations represent different lines that cross at one specific point, showing one unique solution.

Infinitely Many Solutions: Both the right and left sides of the equations are equivalent expressions.

If two equations are the same when simplified, then they represent the same line, leading to an infinite number of intersection points.

Infinitely Many Solutions: 2x + 5 is a multiple of x + 5.

This means that for any value of x, the two expressions are equal at all points provided. Thus, they yield infinitely many solutions.

Zero Solutions: y is equal to two different expressions.

If two expressions for y yield different values for the same x, there can be no point of intersection, resulting in zero solutions.

Zero Solutions: 2x + 5 = x + 5 has no solutions.

Rearranging gives x = 0; however, substituting back shows contradiction, thus confirming no solutions exist.

If you need further assistance with this topic, feel free to check our extended services page!

Adding equations A and B is a valid method for solving a system of equations because it maintains the equality. When you add or subtract the same quantity from both sides of an equation, the equation remains true. This means that if both equations describe the same relationship (the same variables), their sum will also describe a valid relationship.

In your example, since the y-terms cancel during the addition, you simplify the problem, allowing you to isolate x. Thus, this method effectively reduces the complexity of the system without losing any information about the solution. If done correctly, it will lead you to the actual values of the variables, ensuring you get the correct solution, x = 3 and y = 1.

To determine the correct statements about the solution sets for each system of equations, we can analyze each statement:

1. Zero Solutions:

– “4x + 2 can never be equal to 4x – 2.”

– This is true, as simplifying shows that it leads to a contradiction (2 ≠ -2).

2. Zero Solutions:

– “y is equal to two different expressions.”

– This generally means that the two expressions cannot equal each other, resulting in no solution.

3. One Solution:

– “3x – 4 = 2x + 2 has one solution.”

– This can be solved to find that x = 6 is a valid solution, making this statement true.

4. Infinitely Many Solutions:

– “4x + 2 is a multiple of 2x + 1.”

– This suggests a proportional relationship; if true, this can yield infinitely many solutions.

5. Infinitely Many Solutions:

– “Both equations are equal to y.”

– If two equations are both equal to y, they represent the same line, thus having infinitely many solutions wherever they overlap.

### Summary:

– Zero Solutions: “4x + 2 can never be equal to 4x – 2.” & “y is equal to two different expressions.”

– One Solution

The correct answer is: The kayak rentals and the ice-cream purchases have a positive correlation, but neither may cause the other.

Explanation: A positive correlation indicates that as one variable increases, the other variable tends to increase as well. In this case, while there may be a trend where more kayak rentals coincide with more ice-cream purchases, this does not imply that one causes the other. They could both be influenced by a third factor, such as warm weather or a nearby beach location, leading to increased activity in both areas without a direct causal link.

To find the correct equation to determine the dimensions of the original cardboard piece, we can start by setting up the problem.

When Isabelle cuts out 2-inch squares from each corner of the square cardboard, she reduces both dimensions of the cardboard by 4 inches (2 inches from each side). Let ( w ) be the original width (and length, since it’s square) of the cardboard. The dimensions of the base of the box after folding up the sides will be ( (w – 4) ) inches by ( (w – 4) ) inches.

The volume ( V ) of the box can be expressed as:

[ V = text{Base Area} times text{Height} = (w – 4)(w – 4)(2) ]

This can be simplified to:

[ V = 2(w – 4)^2 ]

We know the volume is 98 cubic inches, so we can set up the equation:

[ 2(w – 4)^2 = 98 ]

Thus, the correct answer is 2(w – 4)² = 98. This equation can be used to find the dimensions of the original cardboard piece.

If you have any more questions or need further assistance, feel free to ask!

To determine the inequality that describes Timothy’s deck dimensions, let’s define the width as ( w ). The length of the deck, based on the problem, will be ( 2w – 4 ).

The area of a rectangle (deck) is calculated as length times width, so we can express the area as:

[

text{Area} = w times (2w – 4)

]

We want this area to be at least 140 square feet, so we set up the inequality:

[

w(2w – 4) geq 140

]

This can be rearranged to form:

[

2w^2 – 4w – 140 geq 0

]

However, if we want it in the form of an inequality that can be solved directly, we look for the equivalent form, which is:

[

2w^2 – 4w – 140 leq 0

]

Therefore, the correct answer is:

2w² – 4w – 140 ≤ 0

This inequality can be solved to find possible widths for Timothy’s deck. Great job on working through this problem! If you need more help, don’t hesitate to check the extended services page for additional resources.

To find out how much more brown sugar Sebastian used compared to white sugar, we can subtract the amount of white sugar from the amount of brown sugar.

Sebastian used:

– Brown sugar: ( frac{4}{5} ) scoop

– White sugar: ( frac{1}{2} ) scoop

First, we need a common denominator to subtract these fractions. The least common denominator for 5 and 2 is 10.

Now, convert both fractions:

– Brown sugar: ( frac{4}{5} = frac{8}{10} )

– White sugar: ( frac{1}{2} = frac{5}{10} )

Now, we can subtract the amounts:

[

frac{8}{10} – frac{5}{10} = frac{3}{10}

]

So, Sebastian used ( frac{3}{10} ) scoop more brown sugar than white sugar.

Final Answer: ( frac{3}{10} ) scoop more brown sugar.

To find out how much it rained in the second half of the month, we need to subtract the rainfall in the first half from the total rainfall for the month.

1. Total rainfall for the month: (3 frac{1}{8}) inches
2. Rainfall in the first half: (frac{1}{2}) inch

First, convert (3 frac{1}{8}) to an improper fraction:

[

3 frac{1}{8} = frac{3 times 8 + 1}{8} = frac{24 + 1}{8} = frac{25}{8}

]

Next, convert (frac{1}{2}) to have a common denominator with (frac{25}{8}):

[

frac{1}{2} = frac{4}{8}

]

Now, subtract the rainfall from the total:

[

frac{25}{8} – frac{4}{8} = frac{21}{8}

]

Now, convert (frac{21}{8}) back to a mixed number:

[

frac{21}{8} = 2 frac{5}{8}

]

So, the rainfall in the second half of the month is (2 frac{5}{8}) inches.

Final Answer: (2 frac{5}{8}) inches.

To make a half recipe of 2 2/3 teaspoons of mint, we first convert the mixed number to an improper fraction.

1. Convert 2 2/3 to an improper fraction:

– 2 2/3 = (2 * 3 + 2) / 3 = (6 + 2) / 3 = 8/3 teaspoons.

2. Now, to find half of 8/3 teaspoons:

– (1/2) * (8/3) = 8/6 = 4/3 teaspoons.

So, you would need 4/3 teaspoons of mint for a half recipe. This is the same as 1 1/3 teaspoons if you prefer it as a mixed number.

To find out how much farther the beetle crawled than the spider, we can subtract the distance the spider crawled from the distance the beetle crawled.

1. The beetle crawled 2 yards.
2. The spider crawled ( frac{1}{5} ) of a yard.

Now, we need a common denominator to subtract these. The common denominator for 1 and 5 is 5.

Convert 2 yards:

[

2 = frac{2 times 5}{1 times 5} = frac{10}{5}

]

Now, we can subtract the distances:

[

frac{10}{5} – frac{1}{5} = frac{10 – 1}{5} = frac{9}{5}

]

The beetle crawled ( frac{9}{5} ) yards farther than the spider. This can also be expressed as a mixed number:

[

frac{9}{5} = 1 frac{4}{5} text{ yards}

]

So, the answer is ( frac{9}{5} ) or ( 1 frac{4}{5} ) yards. Great job working through the problem! If you need more practice or deeper explanations, feel free to check the extended services page for additional help.

To find out how much farther away Planet X is from Planet Y, you need to subtract the distance of Planet Y from the distance of Planet X.

1. Distance of Planet X: ( frac{1}{6} ) light-year
2. Distance of Planet Y: ( frac{1}{12} ) light-year

Step 1: Find a common denominator.

The least common multiple of 6 and 12 is 12.

Step 2: Convert the fractions.

– For Planet X:

( frac{1}{6} = frac{2}{12} ) (because ( 1 times 2 = 2 ) and ( 6 times 2 = 12 ))

– For Planet Y:

( frac{1}{12} ) remains ( frac{1}{12} ).

Step 3: Subtract the distances.

Now, subtract the distance of Planet Y from Planet X:

( frac{2}{12} – frac{1}{12} = frac{1}{12} ) light-year.

Conclusion:

Planet X is ( frac{1}{12} ) light-year farther away from Earth than Planet Y.

To determine how many rolls of tape Robert needs to buy, we divide the total length of tape he needs (5 5/8 feet) by the length of tape on each roll (1 7/8 feet).

First, convert the mixed numbers to improper fractions:

1. Convert 5 5/8 to an improper fraction:

– 5 5/8 = (5 × 8 + 5) / 8 = (40 + 5) / 8 = 45/8

2. Convert 1 7/8 to an improper fraction:

– 1 7/8 = (1 × 8 + 7) / 8 = (8 + 7) / 8 = 15/8

Now, divide 45/8 by 15/8:

[

frac{45/8}{15/8} = frac{45}{15} = 3

]

Robert needs 3 rolls of tape.

If Robert needs to ensure he has enough, he should round up (if needed) to ensure he doesn’t run out of tape, but since the calculation shows exactly 3 rolls, that’s his answer.

So, Robert should buy 3 rolls of tape.

The correct conclusion is: The kayak rentals and the ice-cream purchases have a positive correlation, but neither may cause the other.

Explanation: A positive correlation means that as one variable increases, the other variable tends to increase as well. In this case, if both kayak rentals and ice-cream purchases increase together, it suggests a relationship but does not imply that one causes the other. There could be another factor influencing both, such as warmer weather encouraging outdoor activities and ice cream sales. Thus, while they are positively correlated, we cannot confidently say that one causes the other.

If you have more questions or need further clarification, feel free to ask!

To find out how many bales of hay the horses are fed each day, you will multiply the amount the cattle are fed by 1 5/6.

1. The cattle are fed 5 bales of hay.
2. Multiply this by 1 5/6:

– First, convert 1 5/6 to an improper fraction.

1 5/6 = (6 * 1 + 5)/6 = 11/6.
3. Now, multiply:

[

5 times frac{11}{6} = frac{5 times 11}{6} = frac{55}{6}.

]
4. To express (frac{55}{6}) as a mixed number:

– Divide 55 by 6, which equals 9 with a remainder of 1. So, (frac{55}{6} = 9 frac{1}{6}).

Therefore, the horses are fed (9 frac{1}{6}) bales of hay each day.