Problem 29 - Algebra Form and Function

Question

In Problems \(29-33,\) without solving the equations, decide how many solutions the system has. $$ \left\\{\begin{array}{r} x-2 y=7 \\ x+y=9 \end{array}\right. $$

Step-by-Step Solution

Answer

Answer: The given system of linear equations has a unique solution.

Step 1: Identify the given equations

The given system of linear equations is: $$ \left\\{\begin{array}{r} x-2y=7 \\ x+y=9 \end{array}\right. $$

Step 2: Write the given equations in general form

We can write the given system of linear equations in the general form as: $$ \left\\{\begin{array}{r} 1x - 2y = 7 \\ 1x + 1y = 9 \end{array}\right. $$

Step 3: Check the coefficients and constants

Let's denote the coefficients and constants from both equations: For the first equation, \(a_1 = 1\), \(b_1 = -2\), and \(c_1 = 7\). For the second equation, \(a_2 = 1\), \(b_2 = 1\), and \(c_2 = 9\).

Step 4: Analyze the coefficients and constants

Now we will compare the coefficients and constants of the two equations. If \(a_1/a_2 = b_1/b_2 = c_1/c_2\), then the system has infinitely many solutions. If \(a_1/a_2 = b_1/b_2 \neq c_1/c_2\), then the system has no solutions. Otherwise, the system has a unique solution. Calculating the ratios: $$ \frac{a_1}{a_2} = \frac{1}{1} = 1 ; \frac{b_1}{b_2} = \frac{-2}{1} = -2 ; \frac{c_1}{c_2} = \frac{7}{9}; $$ Since \(\frac{a_1}{a_2} \neq \frac{b_1}{b_2}\), we don't need to check the ratio of constants.

Step 5: Determine the number of solutions

Since the ratios do not satisfy the conditions for either infinitely many solutions or no solutions, it can be concluded that the given system of linear equations has a unique solution.

Key Concepts

Unique Solution

A unique solution in a system of linear equations arises when there is exactly one set of values for the variables that satisfy all equations simultaneously. To determine if a system has a unique solution, one can compare its coefficients and constants.
When analyzing a system of two linear equations, as in our example, a unique solution occurs when the conditions for infinitely many or no solutions are not met. These conditions involve the ratios of the coefficients of the variables and the constants.
In this exercise, the ratios of the coefficients are different, specifically

• \(\frac{a_1}{a_2} = 1\)
• \(\frac{b_1}{b_2} = -2\)

This difference indicates that the equations represent lines that intersect at exactly one point. Thus, the system has a unique solution.

Comparing Coefficients

Comparing coefficients is a fundamental step to finding out the number of solutions in a system of equations. By looking at the ratios of coefficients of similar terms in the equations, we get an idea about the geometric relationship of the lines represented by those equations.
Here's a breakdown of how to compare coefficients:

• Identify the coefficients of the corresponding variables and the constants in each equation.
• Calculate the ratios of the coefficients. In our case, these are \(\frac{a_1}{a_2}, \frac{b_1}{b_2},\) and \(\frac{c_1}{c_2}\).
• Analyze these ratios to determine the nature of the system—whether it has one solution, many solutions, or none.

In this example, since \(\frac{a_1}{a_2} = 1\) and \(\frac{b_1}{b_2} = -2\) are not equal, it suggests that the lines intersect at a single point, hence a unique solution.

General Form of Equation

The general form of a linear equation is typically expressed as \(ax + by = c\). This form helps in directly comparing the coefficients \(a\), \(b\), and the constant \(c\) from different equations in a system. For the given system, the equations can be written as:

• \(1x - 2y = 7\)
• \(1x + 1y = 9\)

This form is particularly useful because it allows us to make straight comparisons between different equations.
Converting equations into the general form is the first step in many solution processes. It sets the stage for methods such as substitution, elimination, or the comparison of coefficients.
Writing equations in this way ensures uniformity and simplicity when dealing with systems of equations, making it easier to deduce conclusions directly.