Problem 66 - College Algebra Essentials

Question

Writing in Mathematics Explain how to solve a nonlinear system using the addition method. Use \(x^{2}-y^{2}=5\) and \(3 x^{2}-2 y^{2}=19\) to illustrate your explanation.

Step-by-Step Solution

Answer

The solution to the system of equations is \(x = \pm \sqrt{\frac{39}{5}} \) and \(y = \pm \sqrt{5 - x^{2}} \)

Step 1: Analyze the given equations

In the first instance, the task at hand is to observe the two equations which are \(x^{2}-y^{2}=5\) and \(3 x^{2}-2 y^{2}=19\). The main aim behind this step is to see if these equations can be manipulated or used in a way to find the common term between them.

Step 2: Multiply to get common terms

To find a common term in the two equations, multiply the whole of equation 1 by 3, which will give: \(3x^{2} - 3y^{2} = 15\).

Step 3: Use the addition method

Next step involves the implementation of the addition method. Here, we add equation 1 (from step 2) and equation 2 together. As a result, we get: \((3x^{2} - 3y^{2}) + (3x^{2} - 2y^{2}) = 15 + 19\). This simplifies to \(6x^{2} - 5y^{2} = 34\).

Step 4: Solve for one of the variables

From the equation obtained in Step 3, it is simplified to get: \(y^{2} = 6x^{2} - 34\). Taking square root of both sides to solve for \(y\), we have: \(y = \sqrt{6x^{2} - 34}\)

Step 5: Substitute \(y\) into one of the original equations

Next, we substitute \(y\) into the first original equation. This will be: \(x^{2} - (\sqrt{6x^{2} - 34})^2 = 5 \), this simplifies to \(x^{2} - (6x^{2} - 34) = 5 \), which further simplifies to: \(-5x^{2}+34=5 \). Then we solve for \(x\) to get \(x = \pm \sqrt{\frac{39}{5}} \)

Step 6: Find the corresponding \(y\) values

Finally, substitution of the obtained \(x\)-values into the first original equation to determine the corresponding \(y\)-values. Therefore \(y = \pm \sqrt{5 - x^{2}} \), where \(x = \pm \sqrt{\frac{39}{5}} \)

Key Concepts

Addition Method

The addition method, also known as the elimination method, is a strategic approach for solving systems of equations. This technique involves manipulating equations so that one of the variables can be eliminated when the equations are added together. In such cases, the coefficients of a particular variable become equal in both equations but with opposite signs.
By adding these equations together, that variable cancels out, allowing us to solve for the other variable more easily. This method can be particularly helpful when dealing with nonlinear systems of equations, such as the one involving quadratic terms.

• Start by aligning the equations one below the other.
• Manipulate equations, if needed, to get opposite coefficients for one variable.
• Add the equations to eliminate that chosen variable.

For the given equations, multiplying one equation by a specific number helps us achieve these necessary opposite coefficients. It is a very effective way to make simplification possible and find a solution.

Quadratic Equations

Quadratic equations are polynomial equations of the form \(ax^2 + bx + c = 0\). They can appear in various forms and usually include a variable raised to the second power (\(x^2\)).
The solutions to these equations are found by determining the values of \(x\) that make the equation true. In the context of systems of equations, quadratic equations introduce a nonlinear curve, such as a parabola, into the solution set.
They may have:

• Two real or complex solutions
• One real solution (when both solutions are the same)
• No real solutions (if the discriminant \(b^2-4ac\) is negative)

Understanding how quadratic equations function within a system aids in recognizing patterns and potential solution sets. In nonlinear systems, one might have to graph these quadratic equations to visualize how and where they intersect with other equations.

Solving Systems of Equations

Solving systems of equations involves finding the variable values that satisfy all equations simultaneously. When faced with nonlinear systems, like one with quadratic equations, traditional methods may need adjustment.
Basic approaches include:

• Graphical method - plotting equations to find intersections
• Substitution - solve one equation for one variable and substitute into another
• Addition/elimination method - as seen in this exercise

Each method has its advantages, but the elimination method can often simplify problems involving nonlinear terms. The main challenge lies in handling the complexity that these non-linear relationships introduce, which usually requires extra manipulation of equations, such as multiplying to achieve eliminations or rearranging into simpler forms.
Choosing the right technique depends on the structure of the equations you're working with.

Substitution Method

The substitution method is a commonly used technique for solving systems of equations, particularly useful when at least one equation is easily rearranged to express one variable in terms of the other. This method helps in reducing the number of variables involved in an equation.
Follow these simple steps for substitution:

• Solve one of the equations for one of the variables.
• Substitute the expression into the other equation.
• Solve for the second variable, then backtrack to find the first.

For example, if you solve \(x = 2y + 3\) from one equation, you can substitute it into another equation containing \(x\). This operation simplifies complex systems, especially nonlinear ones, by turning multiple equations into a single equation that's easier to solve.
The process of substitution paired with another method, like elimination, can be particularly powerful in unraveling complex nonlinear systems.