Problem 40 - Algebra and Trigonometry

Question

\(21-48=\) 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

The system has infinitely many solutions, expressed as \((3t+4, t)\).

Step 1: Write the system of equations

Let's denote the equations from the system. The first equation is \(25x - 75y = 100\), and the second equation is \(-10x + 30y = -40\).

Step 2: Simplify the equations

Both equations can be simplified by dividing each term by the greatest common factor. For the first equation, divide every term by 25: \(x - 3y = 4\). For the second equation, divide every term by -10: \(x - 3y = 4\).

Step 3: Compare the simplified equations

After simplification, both equations become \(x - 3y = 4\). This indicates that they are actually the same equation.

Step 4: Determine the nature of the solution

Since the two equations are identical, any point that satisfies the equation \(x - 3y = 4\) is a solution to the system. This means that the system has infinitely many solutions.

Step 5: Expressing the solution

To express the solutions in ordered-pair form, solve the equation \(x = 3y + 4\) for \(x\). Using a parameter \(t\) for \(y\), the solutions can be expressed as ordered pairs: \((3t + 4, t)\), where \(t\) is any real number.

Key Concepts

Infinitely Many Solutions

In the realm of systems of linear equations, a system can have several types of solutions: one unique solution, no solutions, or infinitely many solutions. When a system has infinitely many solutions, it means that there is not just one single solution that satisfies all equations in the system, but an infinite number of them. This usually happens when the system of equations is dependent, meaning the equations are not unique and essentially represent the same mathematical line.
For the given exercise, after simplification, both provided equations turned out to be the same, i.e., they both simplified to \(x - 3y = 4\). As a result, any solution that lies on the line represented by this equation is a solution to the system.

• This indicates the lines overlap completely.
• The equations are dependent and consistent.
• There are infinitely many ordered pair solutions.

Hence, if you find one solution, you can find infinitely many more by parameterizing one of the variables.

Simplification of Equations

Simplification of equations is a crucial step in solving systems of linear equations. This process involves reducing equations to their simplest form, which sometimes can make hidden relationships between equations more apparent. Simplification typically involves dividing all terms of an equation by a common factor, thereby reducing coefficients to their smallest integer form.
In the original exercise, both equations were simplified by their respective greatest common factors:

• In the equation \(25x - 75y = 100\), dividing by 25 yields \(x - 3y = 4\).
• In the equation \(-10x + 30y = -40\), dividing by -10 also results in \(x - 3y = 4\).

After simplification, it became evident that both equations were essentially the same, which directly led to the realization that the system has infinitely many solutions.

Ordered Pairs

Ordered pairs play a vital role when dealing with solutions of systems of equations. An ordered pair denotes a pair of numbers in a specific order, which typically represents a point in a two-dimensional space, such as a coordinate plane.
For the system in the exercise, since there are infinitely many solutions, it is helpful to express these solutions in terms of ordered pairs. To find these pairs, we solve the simplified equation \(x - 3y = 4\) for one of the variables, say \(x\).
By expressing \(x\) in terms of a parameter \(t\) for \(y\), we get \(x = 3t + 4\). This allows you to represent solutions as ordered pairs \((3t + 4, t)\), where \(t\) is any real number.

• This method neatly packages infinitely many solutions according to any chosen value of \(t\).
• It's an efficient way to express solutions as each choice of \(t\) gives a new solution.

This format clearly communicates how the solutions fit together on the line represented by the equation.