Problem 39 - Precalculus Mathematics for Calculus

Question

Solve the system, or show that it has no solution. If the system has infinitely many solutions, express them in the ordered pair form given in Example 6. $$\left\\{\begin{array}{l}6 x+4 y=12 \\\9 x+6 y=18\end{array}\right.$$

Step-by-Step Solution

Answer

The system has infinitely many solutions: \((t, 3 - \frac{3}{2}t)\).

Step 1: Identify the System of Equations

The given system of equations is: \[\begin{align*}6x + 4y &= 12 \9x + 6y &= 18\end{align*}\] We need to determine if this system has one solution, no solution, or infinitely many solutions.

Step 2: Simplify the First Equation

Divide the first equation by 2 to simplify it:\[3x + 2y = 6\] This is equivalent to the original equation but easier to work with.

Step 3: Simplify the Second Equation

Divide the second equation by 3 to simplify:\[3x + 2y = 6\] Notably, this is the same as the simplified first equation.

Step 4: Compare Simplified Equations

Both simplified equations are \(3x + 2y = 6\), which means they are the same line. Therefore, every solution of one equation is a solution to the other.

Step 5: Determine the Nature of the Solutions

Since the two equations represent the same line, the system has infinitely many solutions. These solutions can be expressed in parametric form.

Step 6: Express the Solution Set

To express the solution in ordered pairs, choose a parameter \(t\) and express one variable in terms of the other. Let \(x = t\); substituting into \(3x + 2y = 6\):\[3t + 2y = 6 \implies 2y = 6 - 3t \implies y = 3 - \frac{3}{2}t\]Thus, the solution set is \((t, 3 - \frac{3}{2}t)\) for any real number \(t\).

Key Concepts

Infinitely Many Solutions

When we talk about a system of linear equations having infinitely many solutions, we're really referring to a special situation.
The equations involved describe the same line in a geometric sense. This means that any point on this line will satisfy both equations.
The system described in the exercise represents two identical lines. What happens is that both lines, when plotted on a graph, coincide entirely. Every single point that satisfies the first equation will also satisfy the second. This is unlike having no solution, where the lines would be parallel but distinct, never intersecting. Or having one solution, where the lines would intersect at a single point.

• Infinitely many solutions occur when the system's equations are dependent.
• Every solution of one equation is a solution of the other.
• Geometrically, this means the lines coincide completely.

The main takeaway is that not only is there more than one solution, but there's an entire line of them!

Parametric Form

Expressing solutions in parametric form is a useful strategy when dealing with infinitely many solutions.
It involves introducing a parameter, often denoted as \(t\), to describe the solutions completely.
This parameter gives us flexibility, allowing us to express one variable in terms of another.In this system, the parametric form was used by assigning \(x = t\), a parameter that represents any possible value. By substituting \(x = t\) into the equation, you find the corresponding \(y\) values:

• Start with the simplified equation, like \(3x + 2y = 6\).
• Substitute \(x = t\), yielding \(3t + 2y = 6\).
• Solve for \(y\), resulting in \(y = 3 - \frac{3}{2}t\).
Therefore, every solution can be represented as \((t, 3 - \frac{3}{2}t)\), and \(t\) can be any real number. This is valuable because it expresses all solutions concisely and captures the "infinite" aspect through the parameter \(t\).

Solving Linear Equations

Solving a system of linear equations involves finding all possible sets of values that satisfy all equations simultaneously.
The primary goal is to determine whether the system is consistent (one or infinitely many solutions), or inconsistent (no solutions). There are three main methods often used:

• Substitution: Solve one equation for one variable, then substitute into the others.
• Elimination: Combine equations to cancel out one of the variables.
• Graphical Method: Plot each equation to see where they intersect on a graph.

In the exercise, the elimination method was implicitly used by simplifying and comparing the equations. Each equation was divided by a common factor, revealing they describe the same line. Such techniques are valuable because they help to quickly identify the nature of the solution set. Understanding how these methods apply helps us solve not only simple systems but also complex ones as they arise in real-world applications.