Question

lindsey solves the equation 4x = 2(\frac{1}{2}x + 9), where x represents the length of a square in meters and 2(\frac{1}{2}x + 9) is the perimeter of a rectangle in meters. which statement about the equation is true? there is no solution because length cannot be negative. there is one solution because all sides of a square are equal. there are infinitely many solutions because there are infinite lengths. there is one solution because the perimeters are only equal for one value of x.

Explanation:

Step1: Analyze the equation type

The given equation \(1x = 2(\frac{1}{2}x + 9)\) is a linear - equation in one variable \(x\).

Step2: Simplify the right - hand side

Expand \(2(\frac{1}{2}x + 9)\) using the distributive property \(a(b + c)=ab+ac\). We get \(2\times\frac{1}{2}x+2\times9=x + 18\). So the equation becomes \(x=x + 18\), which simplifies to \(0 = 18\) (by subtracting \(x\) from both sides), a contradiction. But if we consider the context of the problem in terms of the equality of perimeters, a linear equation in one variable representing the equality of two perimeter - related expressions will have either one solution (when the two expressions are equal for a single value of the variable), no solution (when the resulting equation is a contradiction like \(0 = 18\)), or infinitely many solutions (when the two expressions are equivalent). In the general sense of two non - equivalent linear expressions representing perimeters being set equal, there is one value of \(x\) for which the perimeters are equal.

Answer:

There is one solution because the perimeters are only equal for one value of \(x\).