Solve each problem. Which one or more of the following cannot be a correct equation to solve a geometry problem, if x represents the length of a rectangle? (Hint: Solve each equation and consider the solution.) A. 2x+2(x- ) = 14 B. -2x+7(5-x) = 52 C. 5(x+2)+5x = 10 D. 2x+2(x-3) = 22

Verified step by step guidance

1. Step 1: For each equation, solve for \( x \) by isolating \( x \) on one side of the equation.
2. Step 2: For equation A, \( 2x + 2(x - ) = 14 \), distribute and combine like terms to simplify the equation.
3. Step 3: For equation B, \( -2x + 7(5 - x) = 52 \), distribute the 7 and combine like terms to simplify the equation.
4. Step 4: For equation C, \( 5(x + 2) + 5x = 10 \), distribute the 5 and combine like terms to simplify the equation.
5. Step 5: For equation D, \( 2x + 2(x - 3) = 22 \), distribute the 2 and combine like terms to simplify the equation. After solving each equation, check if the solution for \( x \) is a valid length (i.e., \( x > 0 \)).

Key Concepts

Linear Equations

Linear equations are mathematical statements that express the equality of two linear expressions. They typically take the form ax + b = c, where a, b, and c are constants, and x is the variable. Understanding how to manipulate and solve these equations is crucial for determining the possible values of x, especially in geometry problems involving dimensions like length.

Geometry of Rectangles

In geometry, a rectangle is defined by its length and width, with the area calculated as length multiplied by width. When solving equations related to rectangles, it is essential to ensure that the solutions for x (length) are positive, as negative lengths are not physically meaningful. This context helps in evaluating the validity of the equations provided.

Solution Validity

Solution validity refers to the process of checking whether the solutions obtained from an equation make sense within the context of the problem. For instance, in geometry, if a solution yields a negative length for a rectangle, it is deemed invalid. Evaluating the solutions of the given equations against the constraints of the problem is key to identifying which equations are appropriate.