Problem 4 - Algebra and Trigonometry

Question

The following is a system of two linear equations in two variables. $$\left\\{\begin{array}{c}{x+y=1} \\ {2 x+2 y=2}\end{array}\right.$$ The graph of the first equation is the same as the graph of the second equation, so the system has _____ _____ solutions. We express these solutions by writing $$\begin{array}{l}{x=t} \\ {y=}\end{array}$$ where \(t\) is any real number. Some of the solutions of this system are (1,_____),(-3,_____), and (5,_____).

Step-by-Step Solution

Answer

The system has infinitely many solutions. Solutions are expressed as \((t, 1-t)\), with specific solutions being \((1, 0)\), \((-3, 4)\), and \((5, -4)\).

Step 1: Understanding the System Through Simplification

Both equations in the system \( x+y=1 \) and \( 2x+2y=2 \) are essentially stating the same relationship. To see this, simplify the second equation \( 2x + 2y = 2 \) by dividing all terms by 2, resulting in \( x + y = 1 \), which matches the first equation. Therefore, the two equations represent the same line.

Step 2: Determining the Types of Solutions

Since both equations describe the same line, any point \((x, y)\) on this line is a solution. Thus, the system has infinitely many solutions, which can be described by parameterizing one of the variables, typically \( x \) in terms of a parameter \( t \).

Step 3: Expressing the General Solution in Terms of a Parameter

To express the solutions, let \( x = t \), where \( t \) is any real number. Substitute \( t \) for \( x \) in the first equation \( x + y = 1 \), which gives: \( t + y = 1 \). Solving for \( y \): \( y = 1 - t \). Thus, solutions can be expressed as \( (t, 1-t) \).

Step 4: Substituting Values to Find Specific Solutions

Substitute the given values of \( x \) to find corresponding values of \( y \):1. **When \( x = 1 \):** \( y = 1 - 1 = 0 \), so one solution is \( (1, 0) \).2. **When \( x = -3 \):** \( y = 1 - (-3) = 4 \), so another solution is \( (-3, 4) \).3. **When \( x = 5 \):** \( y = 1 - 5 = -4 \), so another solution is \( (5, -4) \).

Key Concepts

Infinitely Many Solutions

In the fascinating world of algebra, when two linear equations in a system effectively describe the same line, they possess infinitely many solutions. This might sound intriguing at first, but it's actually quite intuitive! Let's break it down a bit to ease understanding.

If you picture a line on the Cartesian plane, it consists of countless points. All these points are potential solutions since they satisfy the equation of the line.

• In the system given, the equations \( x + y = 1 \) and \( 2x + 2y = 2 \) simplify to the exact same line.
• Every procedure and solution that applies to one equation applies to the other.

Hence, any point on this line becomes a solution to the system, yielding infinitely many solutions. This means there is not just one point or a few points, but an endless array of points that satisfy both equations simultaneously.

Line Parameterization

When solving systems of equations with infinitely many solutions, parameterization is a powerful tool. It allows us to express the solutions in a clear, generalized form. In our system, parameterization lets us describe the line represented by \( x + y = 1 \) efficiently and comprehensively.

Here's how it works:

• We choose one variable as a parameter, often denoted as \( t \).
• For instance, let \( x = t \) where \( t \) is any real number.
• Substituting \( x = t \) into the equation, we get \( t + y = 1 \).

Solving for \( y \), we find \( y = 1 - t \). Thus, any solution to the system can be expressed as \( (t, 1-t) \).

This parameterization provides us with a simple way to generate specific solutions by substituting different values for \( t \), confirming the infinite nature of the solutions.

Solution Sets in Algebra

Understanding the variety of solution sets is crucial in algebra as it allows us to categorize and solve equations effectively. In this context, when two equations in a system overlap entirely, they define a singular line. This line symbolizes an infinite solution set.

Here's what distinguishes these solution sets:

• Solve for a unique solution: This happens when two lines intersect at a single point.
• No solution exists when lines run parallel but never meet.
• Infinitely many solutions occur when lines coincide entirely.

Our system falls under the third category. By expressing solutions using parameters, we show all the potential combinations that satisfy the equations.

It's like painting a vivid picture of possibilities within the constraints of algebra, turning abstract concepts into something tangible.