Problem 8 - College Algebra

Question

Solve the system of nonlinear equations using substitution. $$ \begin{aligned} y &=x \\ x^{2}+y^{2} &=9 \end{aligned} $$

Step-by-Step Solution

Answer

The system has solutions: \( \left(\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}\right) \) and \( \left(-\frac{3\sqrt{2}}{2}, -\frac{3\sqrt{2}}{2}\right) \).

Step 1: Identify Substitution Expression

In the given system, we have one equation that is already solved for a variable: \( y = x \). This allows us to substitute \( x \) for \( y \) in the other equation.

Step 2: Substitute Expression

Substitute \( y = x \) into the second equation \( x^2 + y^2 = 9 \). This yields: \( x^2 + x^2 = 9 \), which simplifies to \( 2x^2 = 9 \).

Step 3: Solve for \( x \)

Solve the equation \( 2x^2 = 9 \) by dividing both sides by 2, giving \( x^2 = \frac{9}{2} \). Take the square root of both sides to find \( x \), which gives \( x = \pm \sqrt{\frac{9}{2}} \), simplifying further to \( x = \pm \frac{3\sqrt{2}}{2} \).

Step 4: Solve for \( y \)

Since \( y = x \), for both solutions of \( x \), \( y \) will be the same: \( y = \pm \frac{3\sqrt{2}}{2} \).

Step 5: Verify Solutions

Substituting \( (x, y) = \left(\frac{3\sqrt{2}}{2}, \frac{3\sqrt{2}}{2}\right) \) and \( (x, y) = \left(-\frac{3\sqrt{2}}{2}, -\frac{3\sqrt{2}}{2}\right) \) into the original equations confirms that both satisfy \( y = x \) and \( x^2 + y^2 = 9 \).

Key Concepts

Substitution Method

The substitution method is a powerful technique for solving systems of equations, especially when dealing with nonlinear equations. It involves isolating one variable in one of the equations and substituting the resulting expression into the other equation.
This approach helps simplify the system, reducing it to a single equation with one variable, which is often easier to solve. In our example, the equation \( y = x \) is already isolated for \( y \), allowing us to substitute \( y \) with \( x \) in the second equation.
The result is a simpler, single-variable quadratic equation: \( x^2 + x^2 = 9 \) or \( 2x^2 = 9 \). By substituting, we effectively transform the problem into finding the solution to this derived equation, making the problem less complex.

Quadratic Equations

Quadratic equations are equations that can be written in the form \( ax^2 + bx + c = 0 \). In our exercise, after substitution, we obtained the quadratic equation \( 2x^2 = 9 \).
This equation can be solved by first simplifying it to \( x^2 = \frac{9}{2} \) by dividing both sides by 2. Quadratic equations often have two solutions, which is why they are significant in mathematics.
These solutions can be found using different methods, including factoring, completing the square, or using the quadratic formula. In this case, since the equation was simplified to \( x^2 = \frac{9}{2} \), we simply take the square root of both sides to find the solutions for \( x \).

System of Equations

A system of equations is a set of two or more equations with the same set of variables. Solving a system means finding the values for these variables that satisfy all the equations simultaneously.
Our problem involved a system comprising the equations \( y = x \) and \( x^2 + y^2 = 9 \). Solving such a system requires finding the values of \( x \) and \( y \) that fit both conditions.

• Linear Equation: \( y = x \)
• Nonlinear Equation: \( x^2 + y^2 = 9 \)

The main challenge in a system of nonlinear equations is that at least one equation involves a nonlinear relationship, adding complexity to the solution process. The substitution method helps manage this complexity by reducing the number of variables.

Square Roots

The square root is a fundamental mathematical operation used to find a number which, when multiplied by itself, gives the original number. When solving quadratic equations, as seen in the exercise, taking square roots is a common step.
After isolating \( x^2 \) to equal \( \frac{9}{2} \), the next step is to find the square root of both sides of the equation. This results in two values: \( x = \pm \frac{3\sqrt{2}}{2} \).
It's essential to understand that the square root operation may yield two possible values: one positive and one negative, since both satisfy the condition when squared. This reflects the symmetric property of quadratic equations about the x-axis in a graph.