A rectangle has a length of 2x + 5 centimeters and a width of x - 2 centimeters. If the pe… — SAT

A rectangle has a length of $2x + 5$ centimeters and a width of $x - 2$ centimeters. If the perimeter of the rectangle is $42$ centimeters, what is the value of $x$ ?

Answer: 6

Answer

The correct answer is 6. The perimeter of a rectangle is calculated using the formula

P = 2l + 2w

. Substituting the given expressions for length and width yields the linear equation

2(2x + 5) + 2(x - 2) = 42

. Distributing the 2 gives

4x + 10 + 2x - 4 = 42

. Combining like terms on the left side simplifies this to

6x + 6 = 42

. Subtracting 6 from both sides yields

6x = 36

. Finally, dividing both sides by 6 gives the value

x = 6

Step-by-Step Solution

Write the perimeter equation in terms of

x

2(2x + 5) + 2(x - 2) = 42

The perimeter of a rectangle is the sum of all its sides, represented by the formula

P = 2l + 2w

Distribute the 2 into the parentheses.

4x + 10 + 2x - 4 = 42

Applying the distributive property yields

2(2x + 5) = 4x + 10

and

2(x - 2) = 2x - 4

Combine like terms on the left side.

6x + 6 = 42

Grouping the variable terms gives

4x + 2x = 6x

, and grouping the constant terms gives

10 - 4 = 6

Subtract 6 from both sides of the equation.

6x = 36

This isolates the variable term on the left side of the equation.

Divide both sides of the equation by 6.

x = 6

Dividing by the coefficient of

x

solves for the variable.

Key Concept

Solving linear equations in one variable by applying the distributive property and combining like terms.

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