Problem 25 - 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. $$\left\\{\begin{array}{l}x+3 y=5 \\\2 x-y=3\end{array}\right.$$

Step-by-Step Solution

Answer

The system has a unique solution: \((2, 1)\).

Step 1: Set Up the System of Equations

We are given the system of equations: \[\begin{align*} x + 3y &= 5, \ 2x - y &= 3. \end{align*}\] Our goal is to determine if there's a unique solution, no solution, or infinitely many solutions.

Step 2: Use the Substitution or Elimination Method

We are going to use the elimination method to solve the system. Multiply the first equation by 2 to align it with the second equation: \[2(x + 3y) = 2(5),\] yielding \[2x + 6y = 10.\] Now, we have the system: \[\begin{align*} 2x + 6y &= 10, \ 2x - y &= 3. \end{align*}\]

Step 3: Eliminate One Variable

Subtract the second equation \(2x - y = 3\) from the new equation \(2x + 6y = 10\): \[ (2x + 6y) - (2x - y) = 10 - 3. \] Simplifying gives \[ 7y = 7. \]

Step 4: Solve for the Remaining Variable

Now, solve for \(y\) from the equation obtained: \[7y = 7\] gives \[y = 1.\]

Step 5: Substitute Back to Find Other Variable

Substitute \(y = 1\) back into one of the original equations. Using \(x + 3y = 5\), we have \[ x + 3(1) = 5, \] which simplifies to \[ x + 3 = 5. \] Solving for \(x\), we get \[ x = 2. \]

Step 6: State the Solution

The solution to the system of equations is the ordered pair \((x, y) = (2, 1)\).

Key Concepts

Elimination Method

The elimination method is a popular technique used to solve systems of linear equations. It involves manipulating the equations to eliminate one of the variables, allowing you to solve for the other variable more easily. In this particular system:

• \(x + 3y = 5\)
• \(2x - y = 3\)

we opt for elimination by aligning the coefficients. By multiplying the first equation by 2 to harmonize the coefficients of \(x\), we obtain \(2x + 6y = 10\). Now, the system becomes:

• \(2x + 6y = 10\)
• \(2x - y = 3\)

Subtracting these aligned equations eliminates \(x\), producing \(7y = 7\). This step crucially converts two equations involving two variables into one equation involving just one, simplifying the problem.
In essence, the elimination method streamlines complex systems into manageable forms by ingeniously removing variables. It's an efficient way to tackle simultaneous equations when substitution is cumbersome.

Unique Solution

A unique solution in a system of equations occurs when the equations intersect at exactly one point, meaning there's only one valid set of values for the variables that satisfy all equations. When using the elimination method in this exercise, we derived two different equations which, upon simplification, led to the value of \(y = 1\).
This indicated a single solution existed for \(y\). Substituting \(y = 1\) back into one of the original equations (\(x + 3(1) = 5\)) allowed us to solve for \(x\), resulting in \(x = 2\).
Thus, the ordered pair \((x, y) = (2, 1)\) is the unique solution to this system of equations.

Understanding Unique Solutions

The key takeaway is that a unique solution shows where the lines intersect in a geometrical context. It's a point where both equations hold true, neither diverging nor converging into infinitely many solutions, but rather meeting precisely once.

Ordered Pair

An ordered pair is a fundamental concept in systems of equations, typically written as \((x, y)\), where \(x\) and \(y\) represent specific values that satisfy both equations. In our system of equations, once we solved for both variables, we found \((x, y) = (2, 1)\).
This ordered pair indicates the point of intersection of the two lines represented by the equations, and it essentially means that if both equations are plotted on a graph, they will meet at this point.

Importance of Ordered Pairs

• Provides a concise method of indicating a solution set.
• Enables easy verification of solutions by substitution back into the original equations.
• Facilitates understanding of how solutions relate to graphical representations and real-world applications.

Ordered pairs are not just about solving a system; they offer insight into the position and behavior of equations relative to one another, presenting a clear picture of their interaction in coordinate geometry.