Problem 40 - 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: \((x, y) = (4 + 3t, t)\), where \(t\) is any real number.

Step 1: Analyze the System of Equations

The given system of equations can be written as: 1. \( 25x - 75y = 100 \) 2. \( -10x + 30y = -40 \). Our goal is to find values of \(x\) and \(y\) that satisfy both equations or determine if the system has no solution or infinitely many solutions.

Step 2: Simplify the Equations

Simplify both equations by dividing through by their greatest common divisors. 1. For the first equation, divide all terms by 25 to get: \( x - 3y = 4 \). 2. For the second equation, divide all terms by -10 to get: \( x - 3y = 4 \). This shows both equations are actually the same.

Step 3: Identify the Type of Solution

Since both simplified equations, \( x - 3y = 4 \), are identical, there are infinitely many solutions. Each point on the line \( x - 3y = 4 \) is a solution to the system.

Step 4: Express the Solution in Parametric Form

To express the solution as an ordered pair, we can choose one variable as the parameter. Let \( y = t \). Substituting into the equation gives \( x = 4 + 3t \). Thus, the solution in parametric form is \((x, y) = (4 + 3t, t)\). Here, \( t \) can be any real number.

Key Concepts

Infinitely Many Solutions

Sometimes, a system of linear equations does not have a single solution or no solution at all, but rather infinitely many solutions. This happens when the equations describe the same line or plane. In the given exercise, the two equations represent the same line after simplifying. Both simplifications resulted in the exact same equation: \( x - 3y = 4 \). This means every point on this line is a solution to the system.

When a system has infinitely many solutions, it is consistent and dependent. Consistent because solutions exist, and dependent because the equations are not independent (they describe the same geometric object). The infinitely many solutions can be visualized as the entire set of points on the line where the equations overlap.

Parametric Form

A useful way to express infinitely many solutions is in parametric form. This form allows us to represent the solutions using a parameter, which can take any real value. In the exercise, we used a parameter \( t \) to express all possible solutions.

To do this, choose one of the variables to be the parameter (in this case \( y = t \)). Substitute \( t \) into the simplified equation \( x - 3y = 4 \). Solving for \( x \), we get \( x = 4 + 3t \). Thus, the parametric form of the solution is \( (x, y) = (4 + 3t, t) \).

This type of expression clearly shows every possible solution point along the line in terms of \( t \). Each different value of \( t \) gives a different point \((x, y)\).

Linear Dependence

Linear dependence in a system of equations occurs when one equation is a multiple of another, meaning they are not distinct and thus describe the same line or plane. In other words, if you can transform one equation into another by multiplying or dividing by a non-zero constant, the equations are linearly dependent.

In our exercise, both given equations initially looked different: \( 25x - 75y = 100 \) and \( -10x + 30y = -40 \). However, by simplifying both, we found they reduce to the same equation \( x - 3y = 4 \). This indicates these equations are linearly dependent.

A system of equations that is linearly dependent will always have either infinite solutions or no solutions, depending on if they overlap or are parallel without intersecting. In our case, they overlap perfectly, leading to infinitely many solutions.