Problem 58 - College Algebra with Modeling and Visualization

Question

If possible, solve the nonlinear system of equations. $$ \begin{array}{r} 2 x-y=0 \\ 2 x y=4 \end{array} $$

Step-by-Step Solution

Answer

The solutions are \((1, 2)\) and \((-1, -2)\).

Step 1: Simplify the First Equation

The first equation is \(2x - y = 0\). This can be rearranged to express \(y\) in terms of \(x\). Solving for \(y\), we add \(y\) to both sides and then divide by \(2\), giving us \(y = 2x\).

Step 2: Substitute into the Second Equation

Now substitute \(y = 2x\) into the second equation \(2xy = 4\). Replace \(y\) with \(2x\) to get \(2x(2x) = 4\). Simplifying, this becomes \(4x^2 = 4\).

Step 3: Solve for x

Divide both sides by 4 to simplify, resulting in \(x^2 = 1\). Next, solve for \(x\) by taking the square root of both sides, which gives us two possible solutions: \(x = 1\) and \(x = -1\).

Step 4: Find Corresponding Values for y

Using the relation \(y = 2x\) from Step 1, calculate \(y\) for both values of \(x\). If \(x = 1\), then \(y = 2(1) = 2\). If \(x = -1\), then \(y = 2(-1) = -2\).

Step 5: Verify Solutions

Plug each pair \((x, y)\) back into the original equations to ensure they satisfy both. For \((x, y) = (1, 2)\):- First equation: \(2(1) - 2 = 0\), True- Second equation: \(2(1)(2) = 4\), TrueFor \((x, y) = (-1, -2)\):- First equation: \(2(-1) - (-2) = 0\), True- Second equation: \(2(-1)(-2) = 4\), True. Both solutions satisfy the original equations.

Key Concepts

Equation Solving

Solving equations involves finding the values of the variables that satisfy all given conditions simultaneously. In this exercise, we deal with a system of nonlinear equations. These equations are called 'nonlinear' because at least one of them consists of terms that are not linear, such as a product of variables.

Here, the system involves two equations:

• The linear equation: \(2x - y = 0\)
• The nonlinear equation: \(2xy = 4\)

The goal is to find the pairs \((x, y)\) that satisfy both equations. This involves several steps: simplifying equations, using algebraic methods like substitution, and verifying if the solutions are valid.

During this process, each step has its purpose. For example, simplifying equations makes substitution easier. Finding solutions usually means manipulating the equations to isolate one of the variables. This helps us identify potential solutions that work for the entire system.

Substitution Method

The substitution method is a powerful algebraic technique used to solve systems of equations, especially when one of the equations can be easily solved for one variable.

In this exercise, the substitution method is applied in these steps:

• Step 1: Simplify and solve the first equation for \(y\). The equation \(2x - y = 0\) is rearranged to express \(y\) in terms of \(x\), resulting in \(y = 2x\).
• Step 2: Substitute \(y = 2x\) into the second equation \(2xy = 4\). This means replacing \(y\) in the equation, transforming it to \(2x(2x) = 4\) which simplifies to \(4x^2 = 4\).
• Step 3: Solve the simplified equation for \(x\), leading to \(x^2 = 1\). Taking the square root provides \(x = 1\) or \(x = -1\).

By substituting one equation into another, we drastically simplify the system, making it much easier to solve. The substitution method essentially reduces the complexity by converting a two-variable problem into a manifestly solvable single-variable problem.

Verification of Solutions

Checking your solutions is a crucial part of solving equations as it ensures accuracy. After determining potential solutions, verify that they satisfy all original equations.

For this set of equations, we have two solutions:

• \((x, y) = (1, 2)\)
• \((x, y) = (-1, -2)\)

To verify, substitute these pairs back into both original equations: - For \((1, 2)\): - First equation: \(2(1) - 2 = 0\), which holds true. - Second equation: \(2(1)(2) = 4\), also true.
- For \((-1, -2)\): - First equation: \(2(-1) - (-2) = 0\), satisfies the equation. - Second equation: \(2(-1)(-2) = 4\), which is correct.

Both solutions meet the requirements of the original system, confirming their validity. Verification is essential in ensuring each step of your equation-solving process has been correctly executed and that no mistakes have been overlooked.