Problem 23 - 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. $$\left\\{\begin{aligned} 2 x-6 y &=10 \\\\-3 x+9 y &=-15 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has infinitely many solutions: \((3y + 5, y)\) for any real number \(y\).

Step 1: Write Down the System of Equations

We have the following system of linear equations: 1. \(2x - 6y = 10\) 2. \(-3x + 9y = -15\). We need to solve these equations simultaneously.

Step 2: Simplify Each Equation

Divide the entire equation 1 by 2 and equation 2 by -3 to simplify:1. \(x - 3y = 5\) 2. \(x - 3y = 5\). Now observe that both simplified equations are identical.

Step 3: Analyze the System

Since both equations \(x - 3y = 5\) are identical, it indicates that the system has infinitely many solutions.

Step 4: Find the General Solution

Let's express \(x\) in terms of \(y\). From \(x - 3y = 5\), solving for \(x\) gives us \(x = 3y + 5\).

Step 5: Write the Solution in Ordered-Pair Form

The general solution can be expressed in ordered-pair form as \((3y + 5, y)\), where \(y\) is any real number.

Key Concepts

Understanding Systems of Equations

A system of equations is a set of two or more equations involving two or more variables. To find the solution to a system of equations, we seek values for the variables that satisfy all the equations simultaneously.
In this case, the given system consists of two linear equations, which means the solution will be the point(s) where the corresponding lines intersect on a graph. Linear equations can be expressed in the form of \( ax + by = c \), and solve for two variables, often \( x \) and \( y \).
There are different outcomes when solving systems of equations:

• One unique solution - when the lines intersect at a single point.
• Infinitely many solutions - when the lines coincide, or overlap completely.
• No solution - when the lines are parallel.

The solution type depends on the relationship between the equations after simplification.

Understanding Infinitely Many Solutions

When solving a system of equations and discovering that both simplified equations are identical, as in this example, it indicates the system has infinitely many solutions. This means the two lines represented by the equations are the same line, and thus intersect at every point along them.
For the provided system, after simplifying both equations, we ended up with:

• \( x - 3y = 5 \)
• \( x - 3y = 5 \)

Since both equations are identical, any solution that satisfies one equation satisfies the other. Thus, instead of a unique intersection point, we have an entire line of solutions.
The general solution here is expressed by setting one variable as a free parameter. In this case, we express \( x \) in terms of \( y \):

\[ x = 3y + 5 \]

This method allows us to describe every point on the line as a solution, hence the infinitely many solutions.

Expressing Solutions as Ordered-Pairs

An ordered-pair is a way of representing solutions to a system of equations in the form \( (x, y) \). It shows the relationship between \( x \) and \( y \) distinctly.
In problems with infinitely many solutions, expressing the solutions as ordered-pairs gives a clear way to understand the relationships between variables.
From our simplified system, we expressed \( x \) in terms of \( y \):

• \( x = 3y + 5 \)

By choosing any real number for \( y \), we can determine \( x \) and thus form ordered-pairs for solutions:

• If \( y = 0 \), then \( x = 5 \). Thus the ordered-pair is \( (5, 0) \).
• If \( y = 1 \), then \( x = 8 \). Thus the ordered-pair is \( (8, 1) \).

Generally, the solution is expressed as \( (3y + 5, y) \), indicating that for any value of \( y \), there is a corresponding \( x \). This concisely captures all the solutions in a flexible and easily adaptable form.