Problem 22 - 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. $$\left\\{\begin{aligned}-3 x+5 y &=2 \\ 9 x-15 y &=6 \end{aligned}\right.$$

Step-by-Step Solution

Answer

The system has infinitely many solutions given by \((t, \frac{3}{5}t + \frac{2}{5})\).

Step 1: Identify each equation

The system of equations given is: \(-3x + 5y = 2\) (Equation 1) and \(9x - 15y = 6\) (Equation 2).

Step 2: Simplify the second equation

Notice that the second equation, \(9x - 15y = 6\), can be simplified by dividing every term by 3, which gives \(3x - 5y = 2\) (Simplified Equation 2).

Step 3: Compare both equations

After simplification, Simplified Equation 2 becomes \(3x - 5y = 2\). Multiply Equation 1 by \(-1\) to check for consistency: \(3x - 5y = -2\). Now compare this to Simplified Equation 2. By doing this, it becomes evident that \(-3x + 5y = 2\) notably contrasts with \(3x - 5y = 2\). However, the simplified Equation 2 is actually \(-3x + 5y = 2\), meaning they are multiples.

Step 4: Determine dependence and check for infinite solutions

Since Simplified Equation 2 is a scalar multiple of Equation 1, both equations describe the same line. Therefore, the system has infinitely many solutions. Express these solutions in parametric form: let \(x = t\), then from Equation 1, solve for \(y\): \(-3t + 5y = 2\) which gives \(5y = 3t + 2\) leading to \(y = \frac{3}{5}t + \frac{2}{5}\).

Step 5: Express solutions in ordered pair form

The solutions can be expressed as ordered pairs in terms of \(t\): \((t, \frac{3}{5}t + \frac{2}{5})\). This form represents all solutions for different values of \(t\).

Key Concepts

Solutions to Linear Systems

When working with systems of linear equations, our goal is to find the set of values that satisfy all equations in the system simultaneously. A system can have:

• *A unique solution:* where the lines intersect at a single point.
• *No solution:* where the lines are parallel and do not intersect.
• *Infinite solutions:* where the lines coincide and overlap entirely.

To determine the nature of the solution, we often manipulate and compare the given equations. By looking at their coefficients and trying to simplify or transform them, we can identify if one equation is a multiple of another, indicating they are the same line. A simple algebraic technique is to check for linear dependence, which results in recognizing whether the system has one solution, none, or infinitely many.

Parametric Equations

Parametric equations provide a way to express a set of infinite solutions using a variable parameter from which all solutions can be derived. In the case of linear systems, once it is determined that there are infinite solutions (because the equations represent the same line), parameterization can express these solutions succinctly. To set up parametric equations when solving the linear system:

• Pick a variable (often "x") and call it a parameter, usually 't'.
• Solve one of the equations for the other variable in terms of 't'.

This forms a set of equations describing the line. For instance, if we let \( x = t \), and solve for \( y \) such as \( y = \frac{3}{5}t + \frac{2}{5} \), we can express any point on the line as an ordered pair in terms of 't'. This representation makes it easy to see how each point along the line relates to the value of the parameter 't'.

Infinite Solutions

A system of linear equations with infinite solutions occurs when the equations describe the same line. This happens when one equation can be transformed into another by multiplying by a constant value. In mathematical terms, the system is "dependent" and does not have a single intersection point or completely separate pathways (which would indicate no solutions). Here's how you demonstrate infinite solutions:

• Identify if the equations are scalar multiples of one another by simplifying or multiplying one equation to match the other.
• If both equations are identical after such transformations, it confirms infinitely many solutions.

These solutions can then be parameterized to represent them in a compact form. For example, the system \( -3x + 5y = 2 \) and \( 9x - 15y = 6 \) can be simplified to show they represent the same line, indicating infinite solutions. The whole set of solutions can then be expressed as \( (t, \frac{3}{5}t + \frac{2}{5}) \), where 't' is a parameter denoting any real number.