Problem 14 - Algebra 2

Question

Find the exact solution(s) of each system of equations. $$ \begin{array}{l}{4 x+y^{2}=20} \\ {4 x^{2}+y^{2}=100}\end{array} $$

Step-by-Step Solution

Answer

Solutions are \((3, 4)\) and \((3, -4)\).

Step 1: Rewrite First Equation

The first equation is rewritten for easier substitution: \(4x + y^2 = 20\). From this equation, express \(4x\) in terms of \(y^2\): \(4x = 20 - y^2\).

Step 2: Substitute into Second Equation

Substitute \(4x = 20 - y^2\) into the second equation. This gives \((20 - y^2)^2 + y^2 = 100\).

Step 3: Simplify the Substituted Equation

Simplify the equation \((20 - y^2)^2 + y^2 = 100\) to form one equation in terms of \(y^2\): \((20 - y^2)^2 = 100 - y^2\).

Step 4: Solve Quadratic Equation

After expanding \((20 - y^2)^2\) and simplifying, solve for \(y^2\). This yields a quadratic equation in \(y^2\), which we solve to find the values.

Step 5: Back Substitute to Find x Values

Using \(y^2\) values found, substitute back into \(4x = 20 - y^2\) to solve for corresponding \(x\) values.

Step 6: Verify Solutions in Original Equations

Plug the found pairs of \((x, y)\) back into both original equations to ensure they solve the system.

Key Concepts

Quadratic Equations

Quadratic equations are essential components in mathematics and can be identified by their highest exponent of two. They usually appear in the form

• \(ax^2 + bx + c = 0\)

In this particular exercise, the equations involve quadratic terms with both \(x\) and \(y\). Quadratics are solved using different methods such as factoring, using the quadratic formula, or completing the square.
For our system, when substituting, it produced a quadratic form combining different powers of \(y\), which you then needed to solve to find the possible values of \(y\). The solution often results in two possible solutions because of the square term. Understanding quadratics is crucial, as they frequently occur not just in academics but also in real-world problems like physics and engineering.
Getting comfortable with solving these also aids in developing problem-solving skills.

Substitution Method

The substitution method is a strategy used to solve systems of equations where one equation is solved for one variable. You then substitute this expression into the other equation to find values for both variables.
In our exercise, we started by rewriting the first equation to express \(4x\) in terms of \(y^2\) as \(4x = 20 - y^2\). Substituting this into the second equation made it possible to work with a single equation involving only \(y^2\) variables.
Advantages of the substitution method include simplifying complex systems into more manageable forms and reducing them to one-variable equations. This approach makes it easier to isolate specific variables and solve them step-by-step. This method is particularly handy when equations have one clear variable to isolate or when they align neatly for substitution.
As always, practice makes perfect, so try applying substitution to various systems to enhance your skills.

Solution Verification

Solution verification is a critical step when solving a system of equations. It confirms that the solutions found satisfy all the original equations given in the system.
After finding potential solutions through our simplifications, substitution, and solving quadratics, it's imperative to back-check these solutions by substituting them into both original equations to ensure they are true.

• This process helps catch any mistakes or errors that might have occurred during substitution or arithmetic processes.
• It's crucial for confirming the viability and correctness of the found solutions.

In our problem, verifying ensures that the potential solutions are correct and applicable for the original problem. Remember, a solution can appear right during solving, but without verification, there's always a chance for small overlooked errors. Therefore, always make a habit of this essential step in your work.