91Ó°ÊÓ

Algebraic equations are mathematical statements that involve variables and constants. These equations assert equality between two expressions. For example, in the exercise, the equation is given as: \(\sqrt{2 - q} = 2.5\). To solve algebraic equations, you need to find the value of the variable that makes the equation true. Knowing the basic principles of algebra helps in manipulating and simplifying equations to isolate the variable of interest. In our exercise, the goal is to determine the value of \(q\) that satisfies the equation.

Isolating the variable is a key step in solving algebraic equations. The main goal is to get the variable by itself on one side of the equation. In the provided example \(\sqrt{2 - q} = 2.5\), the term \(\sqrt{2 - q}\) containing the variable is already isolated on the left side of the equation. This means we can move directly to the next step without additional simplification. However, in other problems, you might need to perform operations such as addition, subtraction, multiplication, or division to isolate the variable first. Once the variable is isolated, the equation becomes simpler to solve.

Squaring both sides of an equation is a method used to eliminate square roots. In mathematical terms, squaring means to multiply a number by itself. When solving the given equation \(\sqrt{2 - q} = 2.5\), squaring both sides cancels out the square root. Here’s the step-by-step transformation:
\[ (\sqrt{2 - q})^2 = (2.5)^2 \] Simplifies to:
\[ 2 - q = 6.25 \] This process eliminates the square root and converts the original equation into a simpler algebraic form. From there, you can solve for \(q\) by further isolating the variable. It is important to perform the same operation (in this case, squaring) on both sides of the equation to maintain the equality. This ensures the solution is valid.