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Algebraic substitution is a method used to solve a system of equations by substituting one variable in terms of another. This technique simplifies equations, making them easier to solve. In our problem, we have two equations: \(p^2 + 4h^2 = 4\) and \(h = p + 1\). The trick is using the second equation to replace \(h\) in the first equation. Here's why this substitution is crucial:- It reduces the number of variables in one equation from two to one. - This makes the equations simpler to handle, especially when more complex operations like squaring are involved.So, when we substitute \(h = p + 1\) into the first equation, it allows us to manage just one variable, \(p\), making the equation simpler: \(p^2 + 4(p+1)^2 = 4\). This step sets the stage for us to solve the equation comfortably.

A quadratic equation is a second-degree polynomial equation in a single variable, generally of the form \(ax^2 + bx + c = 0\). In our example, after substituting \(h\), the original equation transforms into a quadratic: \(5p^2 + 8p = 0\). To solve this quadratic equation, follow these steps:1. **Recognize the Equation Form:** Ensure you identify the standard quadratic equation form by re-arranging if needed. 2. **Factorization:** Look for common factors or apply factorization methods. Here, we have \(p(5p + 8) = 0\), which breaks down our equation into simpler parts.3. **Solving for Roots:** Each factor gives a possible solution; specifically, \(p = 0\) or solving \(5p + 8 = 0\) leads to \(p = -\frac{8}{5}\).Quadratic equations often have two solutions, representing different scenarios in problems involving systems of equations.

Verifying solutions is a critical step in problem-solving to confirm that the answers satisfy all original equations. This ensures accuracy and completeness of your solutions. Here's how you verify the solutions for the given problem:- Once you have potential solutions from solving the equation, such as \( (0, 1) \) and \((-\frac{8}{5}, -\frac{3}{5})\), substitute them back into the original set of equations.- For \( p = 0, h = 1 \), substitute into \( p^2 + 4h^2 = 4 \). The equation checks out: \(0 + 4(1)^2 = 4\).- For \( p = -\frac{8}{5}, h = -\frac{3}{5} \), the first equation still holds: \((-\frac{8}{5})^2 + 4(-\frac{3}{5})^2 = \frac{64}{25} + \frac{36}{25} = 4\).Both solutions satisfy both equations, confirming their correctness. Verification helps avoid errors that might occur during the solution process.