Problem 25 - Algebra and Trigonometry with Analytic Geometry

Question

1-30: Use the method of substitution to solve the system. $$ \left\\{\begin{aligned} y^{2}-4 x^{2} &=4 \\ 9 y^{2}+16 x^{2} &=140 \end{aligned}\right. $$

Step-by-Step Solution

Answer

The solution set is \((\\sqrt{2}, 2\\sqrt{3}), (\\sqrt{2}, -2\\sqrt{3}), (-\\sqrt{2}, 2\\sqrt{3}), (-\\sqrt{2}, -2\\sqrt{3})\).

Step 1: Express y in terms of x

Starting with the first equation, \( y^{2} - 4x^{2} = 4 \), we can express \( y^{2} \) in terms of \( x \):\[ y^{2} = 4x^{2} + 4 \].

Step 2: Substitute into the second equation

Substitute \( y^{2} = 4x^{2} + 4 \) from the first equation into the second equation: \( 9y^{2} + 16x^{2} = 140 \). - Replace \( y^{2} \) in the second equation: \[ 9(4x^{2} + 4) + 16x^{2} = 140 \]. - This simplifies to: \[ 36x^{2} + 36 + 16x^{2} = 140 \].

Step 3: Simplify and solve for x

Combine like terms:\[ 52x^{2} + 36 = 140 \].Subtract 36 from both sides to solve for \( x^{2} \):\[ 52x^{2} = 104 \].Divide by 52:\[ x^{2} = 2 \].Take the square root:\( x = \sqrt{2} \) or \( x = -\sqrt{2} \).

Step 4: Solve for y using x values

Use the values of \( x \) to find \( y \). Substitute \( x = \sqrt{2} \) into the expression from Step 1:\[ y^{2} = 4(\sqrt{2})^2 + 4 = 8 + 4 = 12 \].Thus, \( y = \sqrt{12} = 2\sqrt{3} \) or \( y = -2\sqrt{3} \).Repeat for \( x = -\sqrt{2} \), and since \( y^{2} \) remains 12, the solutions are the same.

Step 5: State the solution set

Combine the solutions:For \( x = \sqrt{2} \), \( y = 2\sqrt{3} \) or \( y = -2\sqrt{3} \).For \( x = -\sqrt{2} \), \( y = 2\sqrt{3} \) or \( y = -2\sqrt{3} \).The solution set is: \[ (\sqrt{2}, 2\sqrt{3}), (\sqrt{2}, -2\sqrt{3}), (-\sqrt{2}, 2\sqrt{3}), (-\sqrt{2}, -2\sqrt{3}) \].

Key Concepts

method of substitution

The method of substitution is a powerful tool used to solve systems of equations, especially when dealing with algebraic problems. In this method, one of the equations is rearranged to solve for one variable in terms of the other. You then substitute this expression into the other equation(s). This transforms a system of two equations into just one equation with one variable.

Let's see how this works in the given problem: We start with two equations. First, we rearrange the first equation, \( y^{2} - 4x^{2} = 4 \), to express \( y^2 \) in terms of \( x \):

• \( y^{2} = 4x^{2} + 4 \)

We then substitute this expression for \( y^2 \) in the second equation, which allows us to find the value of \( x \). By methodically substituting and simplifying, substitution turns complex systems into manageable ones. Understanding this technique is essential for algebra students as it is widely applicable in more advanced mathematical problems.

algebra

Algebra is the branch of mathematics that deals with symbols and the rules for manipulating these symbols. It's a language of its own used to describe mathematical concepts clearly and succinctly. When solving systems of equations, algebra employs certain techniques, like substitution, to discover unknown values.

In our exercise, algebra helps us manipulate the equations into forms that reveal solutions. Using algebraic principles:

• We express one variable in terms of another.
• We substitute and simplify to isolate variables.
• We handle complex expressions effectively by applying well-established rules.

Overall, mastering algebra is like learning the grammar of mathematics—one that enables clear communication and understanding of intricate mathematical ideas.

solution set

The solution set of a system of equations is the collection of values that satisfy all the equations in the system. In our scenario, the solution set reveals the intersection points of the curves given by the equations. To ensure the solution is correct, each point in the solution set should satisfy both equations.

For the equations:

• When \( x = \sqrt{2} \), \( y = 2\sqrt{3} \) or \( y = -2\sqrt{3} \).
• When \( x = -\sqrt{2} \), \( y = 2\sqrt{3} \) or \( y = -2\sqrt{3} \).

The solution set is:

• \( (\sqrt{2}, 2\sqrt{3}) \)
• \( (\sqrt{2}, -2\sqrt{3}) \)
• \( (-\sqrt{2}, 2\sqrt{3}) \)
• \( (-\sqrt{2}, -2\sqrt{3}) \)

Always verify each tuple to confirm it satisfies the original system of equations. Comprehending the solution set is crucial, as it outlines all possible solutions that resolve the equations, which is often the ultimate goal of solving such mathematical problems.