Problem 25 - Algebra and Trigonometry

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 3. $$\left\\{\begin{array}{l}{6 x+4 y=12} \\ {9 x+6 y=18}\end{array}\right.$$

Step-by-Step Solution

Answer

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

Step 1: Analyze the System of Equations

The system provided is:\[\begin{align*}6x + 4y &= 12, \9x + 6y &= 18.\end{align*}\] The goal is to find if there are any solutions, and if so, determine the number of solutions. If no solution exists or if solutions are infinite, we should express this conclusion accordingly.

Step 2: Simplify the Equations

First, simplify each equation if possible. Dividing all terms in the first equation by 2, we obtain: \ \(3x + 2y = 6\). Similarly, divide all terms in the second equation by 3: \ \(3x + 2y = 6\).

Step 3: Compare the Simplified Equations

After simplifying, both equations in the system are identical: \\[3x + 2y = 6.\] Thus, the system actually represents the same line, meaning any point \((x, y)\) on this line will satisfy both equations.

Step 4: Describe the Solution Set

Since both equations represent the same line, the system has infinitely many solutions. We can express solutions in ordered-pair form by solving for \(y\). Taking the simplified equation \(3x + 2y = 6\), solve for \(y\): \ \(2y = 6 - 3x\) and \(y = 3 - \frac{3}{2}x\). The solutions are \((x, 3 - \frac{3}{2}x)\).

Key Concepts

infinitely many solutions

When discussing systems of linear equations, one common outcome is that the system may have infinitely many solutions. This happens when the equations in the system represent the same line. In such cases, any point on these lines is a solution to the system.
In other words, the equations are dependent, so instead of finding a single intersection point, you find that the lines overlap completely.
This particular scenario is classified as having infinitely many solutions, a term that reflects the infinite number of possible values for the variables that satisfy all equations in the system.

solving linear equations

Solving linear equations involves finding the values of variables that make the equation true. For a system of equations, this often means finding the points where their graphs intersect. There are several methods to solve systems of linear equations, such as substitution, elimination, or graphing.
Each method has its own advantages, and the choice of method can depend on the specific system you are solving. Simplifying equations by dividing or multiplying all terms can also help make the solving process easier and the relationships between the equations clearer.
In the given example, observing the equations and simplifying was key to identify the nature of their solutions.

ordered-pair form

The ordered-pair form is a way to express solutions to systems of equations. An ordered pair, like \((x, y)\), represents a solution where \(x\) and \(y\) are values that satisfy the equation. In systems with infinitely many solutions, expressing solutions as an ordered pair allows us to describe the set of all possible solutions concisely.
For instance, if we solve for \(y\) in terms of \(x\), we can write the solution as \((x, f(x))\). This ordered pair format clearly shows the dependency between the variables and helps visualize the solution set on a graph.

identical equations

In the context of systems of linear equations, identical equations occur when simplifying different equations in the system leads to equations that are exactly the same. This means they represent the same line on a graph.
Identical equations reveal that the system does not have a unique solution or a finite number of solutions. Instead, every point on this line is a solution, indicating an infinite set of solutions. This was evident in the exercise when both simplified equations turned out to be \(3x + 2y = 6\), clearly showing that these equations are identical and thus represent the same line.