Question

A rectangle has a perimeter of 14 feet. Twice its length is equal to 1 less than 4 times its width. Write and solve a system of linear equations.

Answer 1

To solve the problem, set up a system of linear equations using the given information. Solve the system to find the values of length and width.

Step-by-step explanation:

To solve this problem, we need to set up a system of linear equations based on the given information. Let's assume the length of the rectangle is l and the width is w.

From the given information, we know that the perimeter of the rectangle is 14 feet, so we can set up the equation: 2(l + w) = 14.

We also know that twice the length is equal to 1 less than 4 times the width, so we can set up the equation: 2l = 4w - 1.

We now have a system of linear equations: 2(l + w) = 14 and 2l = 4w - 1. We can solve this system of equations to find the values of l and w.

Answer 2

The rectangle has a length of 2.75 feet and a width of 1.875 feet.

Here's how to write and solve a system of linear equations to find the length and width of the rectangle:

1. Define variables:

Let l be the length of the rectangle.

Let w be the width of the rectangle.

2. Write equations based on the given information:

Perimeter equation: The perimeter of a rectangle is the sum of the lengths of all sides. Therefore, the perimeter can be expressed as:

2l + 2w = 14

Width and length relationship: We know that twice the length is equal to 1 less than 4 times the width. We can write this as:

2l = 4w - 1

3. Solve the system of equations:

There are two methods to solve the system:

Method 1: Substitution:

1. Solve the second equation for l: l = (4w - 1) / 2

2. Substitute this expression for l in the first equation: 2 * ((4w - 1) / 2) + 2w = 14

3. Simplify and solve for w: 4w - 1 + 4w = 14

4. Combine like terms: 8w - 1 = 14

5. Add 1 to both sides: 8w = 15

6. Divide both sides by 8: w = 1.875

7. Substitute the value of w back into the second equation to find l: l = 2 * 1.875 - 1 = 2.75

Method 2: Elimination:

1. Multiply the second equation by 2: 4l = 8w - 2

2. Subtract the first equation from the modified second equation: 4l - (2l + 2w) = 8w - 2 - (14)

3. Simplify: 2l - 2w = -16

4. Add 2w to both sides: 2l = -16 + 2w

5. Divide both sides by 2: l = -8 + w

6. Substitute this expression for l in the first equation: 2 * (-8 + w) + 2w = 14

7. Simplify and solve for w: -16 + 2w + 2w = 14

8. Combine like terms: 4w - 16 = 14

9. Add 16 to both sides: 4w = 30

10. Divide both sides by 4: w = 7.5

11. Substitute the value of w back into the equation l = -8 + w to find l: l = -8 + 7.5 = -0.5

Both methods give the same result:

Length (l):** 2.75 feet

Width (w):** 1.875 feet

Complete the question:

The rectangle has a perimeter P of 14 feet, and twice its length l is equal to 1 less than 4 times its width w. Write and solve a system of linear equations to find the length and the width of the rectangle.