Problem 22 - Precalculus: Mathematical 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 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: \((x, y) = (t, \frac{2 + 3t}{5})\).

Step 1: Analyze the System

First, observe the given system of equations: \(-3x + 5y = 2\) and \(9x - 15y = 6\). These are two linear equations with two variables, \(x\) and \(y\). We'll check if they have a unique solution, no solution, or infinitely many solutions.

Step 2: Simplify the Second Equation

Look at the second equation \(9x - 15y = 6\). Divide each term by 3 to simplify it: \(3x - 5y = 2\).

Step 3: Compare Equations

Now, compare the simplified second equation \(3x - 5y = 2\) with the first equation, \(-3x + 5y = 2\). Notice that if you multiply the left side of the simplified equation by \(-1\), you get the left side of the first equation, but the right side remains \(2\), not \(-2\).

Step 4: Infer the Relationship Between Equations

Realize that the equations \(-3x + 5y = 2\) and \(3x - 5y = 2\) suggest parallel lines since one is a negative multiple of the other. Since their right-hand sides are the same, they are actually overlapping lines, meaning infinitely many solutions in the form of a line.

Step 5: Express the Solution in Parametric Form

Solve for \(y\) in terms of \(x\) from one of the equations. Use \(x = t\). Substitute \(x = t\) into \(-3t + 5y = 2\) to get \(5y = 2 + 3t\), so \(y = \frac{2 + 3t}{5}\). Thus, the infinitely many solutions are \((x, y) = (t, \frac{2 + 3t}{5})\) where \(t\) is any real number.

Key Concepts

Linear Equations

Linear equations are equations that describe straight lines when graphed. They take the form of:

• x + y = c
• ax + by = c

These equations have two components: variables and constants. The variables (often x and y) represent unknown values that we try to solve for. The terms involving these variables are called linear terms. The constants are just numbers added to or subtracted from these terms. In our exercise, we dealt with two linear equations: \(-3x + 5y = 2\) and \(9x - 15y = 6\). By simplifying or manipulating these equations, we can discover important relationships or properties, such as whether they intersect at a point, coincide, or never meet.

Infinitely Many Solutions

When a system of linear equations has infinitely many solutions, it means that there isn't just a single pair of values (x, y) that satisfies both equations—there are countless pairs. This is often because the equations describe the same line or set of overlapping lines in a graph.
In other words, every point on this line is a solution.

• Infinite solutions occur when the two equations are equivalent after simplification.
• For example, the two equations provided become similar (or the multiples of each other) after manipulating them accordingly.
• When an equation is multiplied by a constant and gives the same expression as another equation, this implies they lay upon the same line.

In this exercise, by simplifying, we find the two linear equations are essentially the same. Hence, the solutions are infinite.

Parametric Form

The parametric form is a method to express the solutions of a system of equations. It allows us to describe a set of solutions using one or more parameters, often designated as \(t\).
This forms a general expression covering all potential solutions, especially useful when dealing with infinitely many solutions.
For our example, we used \(t\) to express the solution of the system

• We set one variable \(x\) as a parameter \(x = t\).
• We then expressed \(y\) in terms of \(t\) using the simplified equation.
• This provided us with the solution set \((x, y) = (t, \frac{2 + 3t}{5})\), where \(t\) can be any real number.

Such parametric forms are powerful because they give us a concise way to handle infinite solutions, illustrating how each variable relates to others systematically.