Section 4.4 - Solving For A Variable In A Linear Relation

1) Farmer Bill has chickens and horses. He has a total of 30 animals. There are a total of 80 legs. Each chicken has 2 legs and each horse has 4 legs. How many of each type of animal does he have?

20 chickens and 10 horses

15 chickens and 15 horses

10 chickens and 20 horses

30 chickens

We can solve this problem by setting up a system of equations. Let's say the number of chickens is represented by C and the number of horses is represented by H. We know that C + H = 30 (because Farmer Bill has a total of 30 animals) and 2C + 4H = 80 (because the total number of legs is 80). Simplifying the second equation, we get C + 2H = 40. Now we can solve these two equations simultaneously. By subtracting the first equation from the second equation, we get H = 10. Substituting this value back into the first equation, we get C = 20. Therefore, Farmer Bill has 20 chickens and 10 horses.

Explanation

We can solve this problem by setting up a system of equations. Let's say the number of chickens is represented by C and the number of horses is represented by H. We know that C + H = 30 (because Farmer Bill has a total of 30 animals) and 2C + 4H = 80 (because the total number of legs is 80). Simplifying the second equation, we get C + 2H = 40. Now we can solve these two equations simultaneously. By subtracting the first equation from the second equation, we get H = 10. Substituting this value back into the first equation, we get C = 20. Therefore, Farmer Bill has 20 chickens and 10 horses.

2) Consider the equation V = LWH. What inverse operation would you use to solve for W?

Addition

Subtraction

Multiplication

Division

To solve for W in the equation V = LWH, we need to isolate W on one side of the equation. Since W is being multiplied by L and H, the inverse operation to undo this multiplication is division. By dividing both sides of the equation by LH, we can cancel out the LH on the right side and be left with W on its own, solving for W. Therefore, the correct inverse operation to solve for W is division.

Explanation

To solve for W in the equation V = LWH, we need to isolate W on one side of the equation. Since W is being multiplied by L and H, the inverse operation to undo this multiplication is division. By dividing both sides of the equation by LH, we can cancel out the LH on the right side and be left with W on its own, solving for W. Therefore, the correct inverse operation to solve for W is division.

3) Libby has $1.85 in quarters and dimes. She has 11 coins in all. How many of each does she have?

5 dimes, 6 quarters

6 dimes, 5 quarters

2 dimes, 9 quarters

4 dimes, 7 quarters

Libby has a total of 11 coins, and the total value of these coins is $1.85. Let's assume she has x dimes and y quarters. The value of x dimes is 0.10x, and the value of y quarters is 0.25y. Since Libby has a total of 11 coins, we can write the equation x + y = 11. Similarly, the total value of the coins is $1.85, so we can write the equation 0.10x + 0.25y = 1.85. By solving these two equations simultaneously, we find that x = 6 and y = 5, which means Libby has 6 dimes and 5 quarters.

Explanation

Libby has a total of 11 coins, and the total value of these coins is $1.85. Let's assume she has x dimes and y quarters. The value of x dimes is 0.10x, and the value of y quarters is 0.25y. Since Libby has a total of 11 coins, we can write the equation x + y = 11. Similarly, the total value of the coins is $1.85, so we can write the equation 0.10x + 0.25y = 1.85. By solving these two equations simultaneously, we find that x = 6 and y = 5, which means Libby has 6 dimes and 5 quarters.

4) Solve A = (L)(W) for L

W = A/L

A = (L)(W)

L = (A)(W)

L = A/W

The given equation A = (L)(W) can be rearranged to solve for L. By dividing both sides of the equation by W, we get L = A/W. This means that in order to find the value of L, we need to divide the value of A by the value of W.

Explanation

The given equation A = (L)(W) can be rearranged to solve for L. By dividing both sides of the equation by W, we get L = A/W. This means that in order to find the value of L, we need to divide the value of A by the value of W.

5) Consider the equation C = 75 + 50t where C represents the cost for a plumer to do a job that takes t hours. Why might someone want to rearrange the equation so that it is solved for t?

They want to hire a plumber for 10h

They want to know how many hours the plumber could work for a certain cost

They want to know the cost of a job

They want to hire an electrician for t hours

Rearranging the equation to solve for t would allow someone to determine how many hours the plumber could work for a certain cost. By isolating the variable t, they can input a specific cost value into the equation and solve for t, giving them the number of hours the plumber could work for that cost. This would provide them with information on the amount of work they can expect from the plumber within their budget.

Explanation

Rearranging the equation to solve for t would allow someone to determine how many hours the plumber could work for a certain cost. By isolating the variable t, they can input a specific cost value into the equation and solve for t, giving them the number of hours the plumber could work for that cost. This would provide them with information on the amount of work they can expect from the plumber within their budget.

6) Solve P = 2L + 2W for W

W = P - 2L

W = (P - 2L) / 2

W = P - L

W = L

The correct answer is W = (P - 2L) / 2 because in the given equation P = 2L + 2W, we need to solve for W. To isolate W, we can start by subtracting 2L from both sides of the equation, which gives us P - 2L = 2W. Then, we divide both sides of the equation by 2 to solve for W, resulting in (P - 2L) / 2 = W.

Explanation

The correct answer is W = (P - 2L) / 2 because in the given equation P = 2L + 2W, we need to solve for W. To isolate W, we can start by subtracting 2L from both sides of the equation, which gives us P - 2L = 2W. Then, we divide both sides of the equation by 2 to solve for W, resulting in (P - 2L) / 2 = W.

7) Solve 3x + 2y = 6 for y in terms of x

Y = -3/2x + 6

Y = x + 3

Y = -3/2x + 3

Y = -3x + 6

The correct answer is y = -3/2x + 3. This is the correct answer because it is obtained by rearranging the given equation 3x + 2y = 6 to solve for y. By subtracting 3x from both sides of the equation and then dividing by 2, we isolate y and obtain y = -3/2x + 3.

Explanation

The correct answer is y = -3/2x + 3. This is the correct answer because it is obtained by rearranging the given equation 3x + 2y = 6 to solve for y. By subtracting 3x from both sides of the equation and then dividing by 2, we isolate y and obtain y = -3/2x + 3.

8) If you multiply a number x by n and then add m the result is p. What is the first inverse operation you should perform to solve this equation for x?

Multiply by m

Divide by m

Add m

Subtract m

To solve the equation for x, we need to undo the operations that were performed on x. The first inverse operation we should perform is to subtract m. By subtracting m, we can isolate the term that was added to x and undo its effect. This will help us find the original value of x before the multiplication and addition were performed.

Explanation

To solve the equation for x, we need to undo the operations that were performed on x. The first inverse operation we should perform is to subtract m. By subtracting m, we can isolate the term that was added to x and undo its effect. This will help us find the original value of x before the multiplication and addition were performed.

9) Solve 3/4x + 1/2y = 3 for y in terms of x

Y = -3/2x + 6

Y = -3x + 3

Y = -3/2x + 12

Y = 3x + 3

The given equation is 3/4x + 1/2y = 3. To solve for y in terms of x, we need to isolate y on one side of the equation. By subtracting 3/4x from both sides, we get 1/2y = -3/4x + 3. To get rid of the fraction, we can multiply both sides by 2, which gives us y = -3/2x + 6. Therefore, the correct answer is y = -3/2x + 6.

Explanation

The given equation is 3/4x + 1/2y = 3. To solve for y in terms of x, we need to isolate y on one side of the equation. By subtracting 3/4x from both sides, we get 1/2y = -3/4x + 3. To get rid of the fraction, we can multiply both sides by 2, which gives us y = -3/2x + 6. Therefore, the correct answer is y = -3/2x + 6.

10) If the area of a traingle is 15cm^2, and the height is 6cm, what is the base?

5 cm

6 cm

5/2 cm

30 cm

The area of a triangle is calculated by multiplying the base by the height and dividing by 2. In this case, the area is given as 15cm^2 and the height is given as 6cm. To find the base, we can rearrange the formula and solve for it. Base = (2 * Area) / height = (2 * 15cm^2) / 6cm = 30cm / 6cm = 5cm. Therefore, the base of the triangle is 5cm.

Explanation

The area of a triangle is calculated by multiplying the base by the height and dividing by 2. In this case, the area is given as 15cm^2 and the height is given as 6cm. To find the base, we can rearrange the formula and solve for it. Base = (2 * Area) / height = (2 * 15cm^2) / 6cm = 30cm / 6cm = 5cm. Therefore, the base of the triangle is 5cm.