Problem 42 - 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. $$\left\\{\begin{aligned} 25 x-75 y &=100 \\ -10 x+30 y &=-40 \end{aligned}\right.$$

Step-by-Step Solution

Answer

Infinitely many solutions: \( (4 + 3t, t) \) for any real \( t \).

Step 1: Write the System of Equations

The given system of linear equations is: \[ \begin{align*} 25x - 75y &= 100 \ -10x + 30y &= -40 \end{align*} \]

Step 2: Simplify Each Equation

Simplify each equation by dividing them by any common factor they may have. For the first equation, divide by 25: \[ x - 3y = 4 \]For the second equation, divide by -10: \[ x - 3y = 4 \]

Step 3: Analyze the Resulting Equations

After simplification, both equations are: \[ x - 3y = 4 \]These are identical, indicating that both equations represent the same line.

Step 4: Determine the Type of Solutions

Since both equations represent the same line, there are infinitely many solutions. Each solution can be expressed in terms of one variable. Let's express it in ordered-pair form.

Step 5: Express the Solution Set

Choose a parameter for one of the variables, e.g., let \( y = t \). Substitute \( y = t \) into \( x - 3y = 4 \):\[ x - 3t = 4 \rightarrow x = 4 + 3t \]The solutions are of the form \( (4 + 3t, t) \) where \( t \) is any real number.

Key Concepts

Linear Equations

Linear equations form the building blocks of algebra. They describe relationships between two variables, usually in the format of a straight line when graphed on a plane.

• The general form of a linear equation is \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants.
• Both variables \( x \) and \( y \) can be adjusted to find various solutions that satisfy the equation.

Linear equations are used to define a "system of equations" when multiple equations are grouped together. Solving a system means finding all values of \( x \) and \( y \) that satisfy all included equations simultaneously. In the given exercise, simplifying the equations revealed that both describe the same line, which is why they are identical after simplification.

Infinitely Many Solutions

When a system of equations has "infinitely many solutions," it indicates that every point on one line is also a point on another. They completely overlap.

• This happens when the equations are equivalent, meaning one can be transformed into the other through multiplication or division by a constant.
• Such systems represent the same linear pattern, but expressed in different forms.

In the problem, simplifying both equations resulted in identical equations, confirming that the system has infinitely many solutions. This reflects that there is not a single intersection point or no intersection, but rather an infinite number of intersection points along the line they share.

Parameterization

Parameterization is a powerful technique used to express solutions of systems with infinite solutions. It creates a way to describe these solutions using a free variable or parameter.

• This involves selecting one variable to act as the parameter, typically denoted as \( t \).
• Other variables are expressed in terms of this chosen parameter.

In the example, by selecting \( y = t \) as a parameter, it leads to finding that \( x = 4 + 3t \). Thus, the solutions can be written in the form \((4 + 3t, t)\). This means for every real number \( t \), there is a corresponding solution for \( x \). This approach outlines how parameterization simplifies expressing potentially infinite solution sets.