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- 1) = 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: Identify the variable and the context. Here, \(x\) represents the length of a rectangle, so \(x\) must be a positive number (greater than zero) because lengths cannot be negative or zero.
2. Step 2: For each equation, isolate \(x\) by simplifying and solving the equation step-by-step. For example, start by expanding parentheses and combining like terms.
3. Step 3: After simplifying, solve for \(x\) by isolating it on one side of the equation. This may involve adding, subtracting, multiplying, or dividing both sides of the equation.
4. Step 4: Once you find the value(s) of \(x\) for each equation, check if the solution is positive. If \(x\) is zero or negative, that equation cannot represent a valid length of a rectangle and thus cannot be a correct equation for the problem.
5. Step 5: Summarize which equations yield valid positive solutions for \(x\) and which do not, based on the solutions found in the previous step.

Key Concepts

Formulating and Solving Linear Equations

Linear equations involve variables raised only to the first power and can be solved using algebraic operations like addition, subtraction, multiplication, division, and distribution. Understanding how to isolate the variable and solve for its value is essential to determine if the equation yields a valid solution for the problem.

Interpreting Solutions in Context

After solving an equation, it is important to interpret the solution within the problem's context. For geometry problems involving lengths, solutions must be positive and make sense physically. Negative or non-real values indicate that the equation cannot represent a valid geometric measurement.

Distributive Property and Simplification

The distributive property allows multiplication over addition or subtraction inside parentheses, e.g., a(b + c) = ab + ac. Proper use of this property is crucial to simplify equations correctly before solving. Misapplication can lead to incorrect equations or solutions that do not fit the problem.