Problem 4 - Precalculus: Mathematics for Calculus

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 $$\begin{array}{l} x=t \\ y=\text{_____} \end{array}$$ where \(t\) is any real number. Some of the solutions of this system are (1,___),(-3,___), and (5,___).

Step-by-Step Solution

Answer

infinitely many; y=1-t

Step 1 – Identify the Equations

The given system of equations is: \( x + y = 1 \) and \( 2x + 2y = 2 \). We need to analyze these equations to determine their relationship.

Step 2 – Simplify the Second Equation

Examine the second equation. It is given as \( 2x + 2y = 2 \). Divide the entire equation by 2 to simplify: \( x + y = 1 \).

Step 3 – Analyze the Simplified System

After simplification, both equations are \( x + y = 1 \). Since these equations are identical, they represent the same line in the coordinate plane.

Step 4 – Determine the Number of Solutions

As both equations represent the same line, there are infinitely many solutions. Any point that satisfies \( x + y = 1 \) is a solution to this system.

Step 5 – Express the Solutions

We express the solutions parametrically: Let \( x = t \), then \( y = 1 - t \), where \( t \) is any real number.

Step 6 – Find Specific Solutions

Using the expression from Step 5, find specific solutions: For \( t=1 \), \((x, y) = (1, 0)\). For \( t=-3 \), \((x, y) = (-3, 4)\). For \( t=5 \), \((x, y) = (5, -4)\).

Key Concepts

System of Equations

A system of equations is a set of two or more equations with the same variables. In our example, we have the equations:

• \( x + y = 1 \)
• \( 2x + 2y = 2 \)

These equations are analyzed together to find the points, or pairs \((x, y)\), that satisfy both equations simultaneously. Essentially, solving a system of equations means finding where the equations "intersect" in terms of solutions that satisfy all the equations in the system. In this case, both equations simplify to the same line, which is crucial to understanding the behavior of their solution set.

Infinite Solutions

When we say a system has infinite solutions, we're talking about having countless pairs \((x, y)\) that satisfy both equations. In our problem, we started with two equations that turned out to be equivalent after simplification. This means every point on the line described by \(x + y = 1\) works for both equations.

Here's why:

• Since both equations are the same line, every point on that line is a solution.
• That line extends infinitely in both directions.
• Every value of \(x\) gives a corresponding \(y\) such that the equation is true.

So, when two equations describe the same line, they share every point on that line, leading to infinite solutions.

Parametric Equations

To effectively express an infinite number of solutions neatly, we use parametric equations. Parametric equations provide a way to describe each element of a solution set with a parameter, often noted as \(t\).

In our system:

• Let \( x = t \). The parameter \( t \) can be any real number, allowing flexibility in choosing specific solutions.
• Then \( y \) becomes \( 1 - t \), utilizing the equation \( x + y = 1 \).

Thus, for any real number \( t \), the pair \((t, 1-t)\) is a solution to the system. This simple association makes it easy to list specific solutions, validate them, and understand how they lie on the line described by the equations.