Problem 64 - College Algebra with Modeling and Visualization

Question

If possible, solve the nonlinear system of equations. $$ \begin{array}{rr} -6 \sqrt{x}+2 y= & -3 \\ 2 \sqrt{x}-\frac{2}{3} y= & 1 \end{array} $$

Step-by-Step Solution

Answer

The system has infinitely many solutions as parameters \(x\) and \(y\) satisfy \(y = 3\sqrt{x} - \frac{3}{2}\).

Step 1: Express Variables Independently

We start by expressing one of the variables in terms of the other from the first equation:\[-6\sqrt{x} + 2y = -3\]. From this equation, express \(y\) as:\[2y = 6\sqrt{x} - 3 \Rightarrow y = 3\sqrt{x} - \frac{3}{2}\].

Step 2: Substitution into the Second Equation

Substitute \(y = 3\sqrt{x} - \frac{3}{2}\) from Step 1 into the second equation \[2\sqrt{x} - \frac{2}{3}y = 1\]. This gives:\[2\sqrt{x} - \frac{2}{3}\left(3\sqrt{x} - \frac{3}{2}\right) = 1\].

Step 3: Simplify the Equation

Simplify the equation from Step 2:\[2\sqrt{x} - 2\sqrt{x} + 1 = 1\]. This simplifies to:\[1 = 1\], which is a true statement but does not involve \(x\).

Step 4: Analyze the Results

Since the equation simplifies to \(1 = 1\), this indicates that the system has infinitely many solutions dependent on the relationship between \(x\) and \(y\). Substituting back \(y = 3\sqrt{x} - \frac{3}{2}\) gives all conditions satisfied simultaneously.

Key Concepts

System of Equations

A **system of equations** involves solving for multiple variables at the same time. Here, we have two equations working together: \(-6\sqrt{x} + 2y = -3\) and \(2\sqrt{x} - \frac{2}{3}y = 1\).

When you solve a system like this one, you're looking for values of \(x\) and \(y\) that work in both equations simultaneously.

There are several methods to solve systems of equations, such as:

• Substitution
• Elimination
• Graphical methods

In our problem, we used substitution. This involves expressing one variable in terms of another using one equation, and then using that expression in another equation. This method is great when dealing with nonlinear equations, like those with square roots.

Recognizing that these equations are not straightforward linear equations imposes requiring creative manipulation through such methods. Each step helps to simplify and eventually solve for the variables while maintaining the balance of the original system.

Sqrt Function

The **sqrt function**, short for square root, is crucial when dealing with nonlinear systems. The square root of a number \(x\), denoted as \(\sqrt{x}\), is a value that, when multiplied by itself, gives \(x\) back.

However, equations with \(\sqrt{x}\) can be tricky because they introduce non-linearity. Meaning not all algebraic rules for simple lines apply, as they can curve or shift differently. Often it is isolated when solving, so we can directly interpret the variable, like \(x\), from an equation.

In the given equations

• \(-6\sqrt{x}\) appears in the first equation
• \(2\sqrt{x}\) appears in the second equation

The presence of \(\sqrt{x}\) requires careful manipulation, such as expressing \(y\) in terms of \(\sqrt{x}\) to help solve. Breaking complex parts into simpler relationships, like expressing one variable in terms of another, is useful in managing these non-linear increments.

Solving Equations

**Solving equations** generally aims at finding values for the variables that make them true, meaning both sides of the equation are equal. Understanding the process requires identifying relationships and breaking complex steps.

In our nonlinear system: 1. We first isolated one variable in terms of the other, \(y = 3\sqrt{x} - \frac{3}{2}\).2. Substituting this back into another equation allowed us to express all parts in terms of \(\sqrt{x}\). This substitution simplifies the system.3. In the last steps, by simplifying the remaining equation, we ended up with a statement \(1 = 1\) which is universally true.

This suggests that instead of a single solution, we have infinitely many solutions. These solutions depend on maintaining a specific relationship between \(x\) and \(y\).

This concept is essential especially when realizing solutions go beyond a particular answer but involve a set or range under certain conditions.