Problem 61 - Intermediate Algebra

Question

Solve each system of equations for real values of \(x\) and \(y.\) $$ \left\\{\begin{array}{l} x^{2}=4-y \\ y=x^{2}+2 \end{array}\right. $$

Step-by-Step Solution

Answer

The solutions are \((x, y) = (1, 3)\) and \((-1, 3)\).

Step 1: Substitute and Set Equations Equal

We start by substituting the expression for \(y\) from the second equation into the first equation. This gives us: \[x^2 = 4 - (x^2 + 2)\] Simplifying the equation, we find: \[x^2 = 4 - x^2 - 2\] \[x^2 = 2 - x^2\]

Step 2: Combine Like Terms

Next, we combine like terms from the equation derived in Step 1: \[2x^2 = 2\] Divide both sides by 2 to isolate \(x^2\): \[x^2 = 1\]

Step 3: Solve for x

The equation \(x^2 = 1\) can be solved to find \(x\): \[x = \pm 1\] Therefore, \(x\) can be either 1 or -1.

Step 4: Solve for y using x values

We use the values of \(x\) (1 and -1) to solve for \(y\). - If \(x = 1\), substitute into the equation \(y = x^2 + 2\): \[y = 1^2 + 2 = 3\]- If \(x = -1\), substitute into the same equation: \[y = (-1)^2 + 2 = 3\]In both cases, \(y = 3\).

Step 5: Verify Solutions

We must verify the solutions with the original system. - For \((x, y) = (1, 3)\), check:1. \(x^2 = 4 - y\) becomes \(1 = 1\), which is true.2. \(y = x^2 + 2\) becomes \(3 = 3\), which is true.- For \((x, y) = (-1, 3)\), check:1. \(x^2 = 4 - y\) becomes \(1 = 1\), which is true.2. \(y = x^2 + 2\) becomes \(3 = 3\), which is true.Both pairs satisfy the system.

Key Concepts

Substitution Method

The substitution method is an effective algebraic technique used to solve systems of equations. This approach involves substituting one equation into another based on the variable that is explicitly defined. Doing so simplifies the system into a single equation with one variable. This method is most helpful when one of the equations is already solved for one variable, allowing easy substitution.
For the given problem, we have two equations as follows:

• Equation 1: \(x^{2} = 4 - y\)
• Equation 2: \(y = x^{2} + 2\)

Instead of solving them simultaneously, we substitute the value of \(y\) from the second equation into the first equation. This simplifies our task, transforming a two-variable system into a quadratic equation. Such conversion makes it easier to solve for one variable at a time. Once \(x\) is determined, we can substitute back to find the second variable, \(y\). Employing the substitution method offers a structured path to reach a solution, particularly useful in solving quadratic systems.

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\), where the highest degree of the variable is 2. In our exercise, after substitution, we obtained a quadratic equation \(x^2 = 1\).
Quadratics can be solved using different methods, such as:

• Factoring
• Completing the square
• Quadratic formula: \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\)

In this instance, simplifying to \(x^2 = 1\) allows us to solve by taking the square root of both sides. This yields two possible real solutions, \(x = 1\) and \(x = -1\).
Quadratic equations can have zero, one, or two solutions depending on the discriminant \(b^2 - 4ac\). A positive discriminant indicates two real solutions, zero discriminant means one real solution, and a negative discriminant suggests no real solutions in the real number system.

Real Solutions

When solving systems of linear and nonlinear equations like in this example, finding real solutions means identifying pairs \((x, y)\) that satisfy all given equations in the set of real numbers.
For this exercise, after isolating \(x\) and subsequently finding \(y\), both pairs \((x,y) = (1,3)\) and \((-1,3)\) emerged as solutions. Both values of \(x\) needed to be tested with the original system to ensure they did not introduce any contradictions. For each pair, verification showed the equality held true for both system equations, confirming they are indeed real solutions.
Real solutions are vital in practical applications where results are expected to logically fit within real-world contexts or scenarios. When solving systems of equations, often multiple tests and verifications are necessary to ensure the integrity and correctness of the solutions found.