Question

The length of a rectangle is 9.7 cm more than 4 times the width. If the perimeter of the rectangle is 91.4 cm, what are its dimensions?

Answer 1

[AI-generated answer — Bot GPT 3.5]

Let the width of the rectangle be x cm.

The length of the rectangle is 9.7 cm more than 4 times the width, so it is (4x + 9.7) cm.
The perimeter of a rectangle is given by the formula P = 2(l + w), where P is the perimeter, l is the length, and w is the width.
Plugging in the given values, we get 91.4 = 2((4x + 9.7) + x).
Expanding the equation, we get 91.4 = 2(5x + 9.7).
Simplifying further, we get 91.4 = 10x + 19.4.
Subtracting 19.4 from both sides, we get 72 = 10x.
Dividing both sides by 10, we get x = 7.2.
So, the width of the rectangle is 7.2 cm.
The length of the rectangle is 4(7.2) + 9.7 = 28.8 + 9.7 = 38.5 cm.
Therefore, the dimensions of the rectangle are 7.2 cm by 38.5 cm.

Answer 2

Solve the system using elimination.5x + 4y = 123x – 3y = 18

Answer 3

[AI-generated answer — Bot GPT 3.5]

To solve the system using elimination, we'll multiply the second equation by 5 to make the coefficients of x in both equations equal:

5(3x - 3y) = 5(18)
15x - 15y = 90

Now, we can subtract the first equation from the second equation to eliminate x:

(15x - 15y) - (5x + 4y) = 90 - 12
15x - 5x - 15y - 4y = 78
10x - 19y = 78

Now, we have a new equation: 10x - 19y = 78.

Multiplying the first equation by 3, we get:

3(5x + 4y) = 3(123)
15x + 12y = 369

We can subtract the equation 10x - 19y = 78 from the equation 15x + 12y = 369 to eliminate x again:

(15x + 12y) - (10x - 19y) = 369 - 78
15x - 10x + 12y + 19y = 291
5x + 31y = 291

Now, we have a new equation: 5x + 31y = 291.

We now have a system of equations:

10x - 19y = 78
5x + 31y = 291

We can multiply the first equation by 5 and the second equation by 10 to further eliminate x:

5(10x - 19y) = 5(78)
50x - 95y = 390

10(5x + 31y) = 10(291)
50x + 310y = 2910

Now, we can subtract the equation 50x - 95y = 390 from the equation 50x + 310y = 2910 to eliminate x once more:

(50x + 310y) - (50x - 95y) = 2910 - 390
50x - 50x + 310y + 95y = 2520
405y = 2520

Dividing both sides by 405, we get:

y = 2520/405
y = 6.22

Now that we have the value of y, we can substitute it back into one of the original equations to solve for x. Let's use the second equation:

3x - 3(6.22) = 18
3x - 18.66 = 18
3x = 18 + 18.66
3x = 36.66
x = 36.66/3
x = 12.22

Therefore, the solution to the system of equations is x = 12.22 and y = 6.22.

Answer 4

Solve the system by substitution.{−4.5x−2y=−12.53.25x−y=−0.75 show little work

Answer 5

[AI-generated answer — Bot GPT 3.5]

From the first equation, we have -4.5x - 2y = -12.5.

Solving for x, we get x = (12.5 - 2y) / -4.5.

Substituting this value of x into the second equation, we have:
3.25[(12.5 - 2y) / -4.5] - y = -0.75.

Now we can solve for y:
3.25(12.5 - 2y) - 4.5y = -0.75.
40.625 - 6.5y - 4.5y = -0.75.
40.625 - 11y = -0.75.
-11y = -0.75 - 40.625.
-11y = -41.375.
Dividing both sides by -11, we get:
y = -41.375 / -11.
y ≈ 3.761.

Substituting this value of y back into x = (12.5 - 2y) / -4.5, we have:
x = (12.5 - 2(3.761)) / -4.5.
x = (12.5 - 7.522) / -4.5.
x = 4.978 / -4.5.
x ≈ -1.106.

Therefore, the solution to the system of equations is x ≈ -1.106 and y ≈ 3.761.

Answer 6

[AI-generated answer — Step-by-Step Bot]

Let's assume the width of the rectangle is "w" cm.

According to the given information, the length of the rectangle is 9.7 cm more than 4 times the width, which means:

Length = 4w + 9.7

The formula to calculate the perimeter of a rectangle is:

Perimeter = 2(Length + Width)

Substituting the given values:

91.4 = 2((4w + 9.7) + w)

Now, let's solve this equation step-by-step:

First, distribute the 2:

91.4 = (8w + 19.4) + 2w

Simplify the equation:

91.4 = 8w + 19.4 + 2w

Combine like terms:

91.4 = 10w + 19.4

Next, subtract 19.4 from both sides of the equation:

91.4 - 19.4 = 10w + 19.4 - 19.4

72 = 10w

Divide both sides of the equation by 10:

72/10 = 10w/10

7.2 = w

Therefore, the width of the rectangle is 7.2 cm.

To find the length, substitute this value back into the equation for the length:

Length = 4w + 9.7
Length = 4(7.2) + 9.7
Length = 28.8 + 9.7
Length = 38.5 cm

So, the dimensions of the rectangle are width = 7.2 cm and length = 38.5 cm.

Answer 7

[AI-generated answer — Explain Bot]

To find the dimensions of the rectangle, we need to set up equations based on the given information and solve them.

Let's assume that the width of the rectangle is "w" cm.

According to the problem, the length of the rectangle is 9.7 cm more than 4 times the width, which can be expressed as:
Length = 4w + 9.7

We also know that the perimeter of a rectangle is given by:
Perimeter = 2(Length + Width)

Substituting the values from above, we can write the equation for the perimeter:
91.4 = 2((4w + 9.7) + w)

Simplifying the equation by removing parentheses:
91.4 = 2(5w + 9.7)

Next, distribute the 2 into the parentheses:
91.4 = 10w + 19.4

Now, isolate the "w" term by subtracting 19.4 from both sides of the equation:
91.4 - 19.4 = 10w
72 = 10w

Finally, divide both sides of the equation by 10 to solve for w:
w = 72 / 10
w = 7.2

So, the width of the rectangle is 7.2 cm.

To find the length, substitute the value of width (w) into the equation for length:
Length = 4w + 9.7
Length = 4(7.2) + 9.7
Length = 28.8 + 9.7
Length = 38.5

Therefore, the dimensions of the rectangle are width = 7.2 cm and length = 38.5 cm.