Problem 64 - Intermediate Algebra

Question

There is another way of telling whether a system of two linear equations in two unknowns is consistent or inconsistent, or whether the equations are dependent, without taking the time to graph each equation. It can be shown that any system of the form $$ \begin{aligned} &a_{1} x+b_{1} y=c_{1} \\ &a_{2} x+b_{2} y=c_{2} \end{aligned} $$ has one and only one solution if $$ \frac{a_{1}}{a_{2}} \neq \frac{b_{1}}{b_{2}} $$ that it has no solution if $$ \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}} \neq \frac{c_{1}}{c_{2}} $$ and that it has infinitely many solutions if $$ \frac{a_{1}}{a_{2}}=\frac{b_{1}}{b_{2}}=\frac{c_{1}}{c_{2}} $$ For each of the following systems, determine whether the system is consistent, the system is inconsistent, or the equations are dependent. (a) \(\left(\begin{array}{l}4 x-3 y=7 \\ 9 x+2 y=5\end{array}\right)\) (b) \(\left(\begin{array}{rl}5 x-y & =6 \\ 10 x-2 y & =19\end{array}\right)\) (c) \(\left(\begin{array}{l}5 x-4 y=11 \\ 4 x+5 y=12\end{array}\right)\) (d) \(\left(\begin{array}{l}x+2 y=5 \\ x-2 y=9\end{array}\right)\) (e) \(\left(\begin{array}{rl}x-3 y & =5 \\ 3 x-9 y & =15\end{array}\right)\) (f) \(\left(\begin{array}{rl}4 x+3 y & =7 \\ 2 x-y & =10\end{array}\right)\) (g) \(\left(\begin{array}{l}3 x+2 y=4 \\ y=-\frac{3}{2} x-1\end{array}\right)\) (h) \(\left(\begin{array}{l}y=\frac{4}{3} x-2 \\ 4 x-3 y=6\end{array}\right)\)

Step-by-Step Solution

Answer

Systems (a), (c), (d), and (f) have one solution each; (b) and (g) are inconsistent; (e) and (h) are dependent.

Step 1: Identify Coefficients for System (a)

The given system is \( 4x - 3y = 7 \) and \( 9x + 2y = 5 \). Here, \( a_1 = 4, b_1 = -3, c_1 = 7 \) and \( a_2 = 9, b_2 = 2, c_2 = 5 \).

Step 2: Determine Ratios for System (a)

Calculate \( \frac{a_1}{a_2} = \frac{4}{9}, \frac{b_1}{b_2} = \frac{-3}{2}, \frac{c_1}{c_2} = \frac{7}{5} \). Since \( \frac{a_1}{a_2} eq \frac{b_1}{b_2} \), the system has one solution.

Step 3: Identify Coefficients for System (b)

The system is \( 5x - y = 6 \) and \( 10x - 2y = 19 \). Here, \( a_1 = 5, b_1 = -1, c_1 = 6 \) and \( a_2 = 10, b_2 = -2, c_2 = 19 \).

Step 4: Determine Ratios for System (b)

Calculate \( \frac{a_1}{a_2} = \frac{5}{10} = \frac{1}{2} \) and \( \frac{b_1}{b_2} = \frac{-1}{-2} = \frac{1}{2} \). But \( \frac{c_1}{c_2} = \frac{6}{19} eq \frac{1}{2} \), so the system is inconsistent.

Step 5: Identify Coefficients for System (c)

The system is \( 5x - 4y = 11 \) and \( 4x + 5y = 12 \). Here, \( a_1 = 5, b_1 = -4, c_1 = 11 \) and \( a_2 = 4, b_2 = 5, c_2 = 12 \).

Step 6: Determine Ratios for System (c)

Calculate \( \frac{a_1}{a_2} = \frac{5}{4} \) and \( \frac{b_1}{b_2} = \frac{-4}{5} \). These are not equal, so the system has one solution.

Step 7: Identify Coefficients for System (d)

The system is \( x + 2y = 5 \) and \( x - 2y = 9 \). Here, \( a_1 = 1, b_1 = 2, c_1 = 5 \) and \( a_2 = 1, b_2 = -2, c_2 = 9 \).

Step 8: Determine Ratios for System (d)

Calculate \( \frac{a_1}{a_2} = \frac{1}{1} = 1 \) and \( \frac{b_1}{b_2} = \frac{2}{-2} = -1 \). These are not equal, so the system has one solution.

Step 9: Identify Coefficients for System (e)

The system is \( x - 3y = 5 \) and \( 3x - 9y = 15 \). Here, \( a_1 = 1, b_1 = -3, c_1 = 5 \) and \( a_2 = 3, b_2 = -9, c_2 = 15 \).

Step 10: Determine Ratios for System (e)

Calculate \( \frac{a_1}{a_2} = \frac{1}{3} \), \( \frac{b_1}{b_2} = \frac{-3}{-9} = \frac{1}{3} \), \( \frac{c_1}{c_2} = \frac{5}{15} = \frac{1}{3} \). All are equal, so the equations are dependent.

Step 11: Identify Coefficients for System (f)

The system is \( 4x + 3y = 7 \) and \( 2x - y = 10 \). Here, \( a_1 = 4, b_1 = 3, c_1 = 7 \) and \( a_2 = 2, b_2 = -1, c_2 = 10 \).

Step 12: Determine Ratios for System (f)

Calculate \( \frac{a_1}{a_2} = \frac{4}{2} = 2 \) and \( \frac{b_1}{b_2} = \frac{3}{-1} = -3 \). These are not equal, so the system has one solution.

Step 13: Identify Coefficients for System (g)

The system is \( 3x + 2y = 4 \) and \( y = -\frac{3}{2}x - 1 \). Substitute \( y = -\frac{3}{2}x - 1 \) into the form \( ax + by = c \) to get \( \frac{3}{2}x + y = -1 \), \( a_2 = \frac{3}{2}, b_2 = 1, c_2 = -1 \).

Step 14: Determine Ratios for System (g)

Calculate \( \frac{a_1}{a_2} = \frac{3}{\frac{3}{2}} = 2 \), \( \frac{b_1}{b_2} = \frac{2}{1} = 2 \). But since \( \frac{c_1}{c_2} = \frac{4}{-1} = -4 \), the system is inconsistent.

Step 15: Identify Coefficients for System (h)

The system is \( y = \frac{4}{3}x - 2 \) and \( 4x - 3y = 6 \). Substitute \( y = \frac{4}{3}x - 2 \) into the form \( ax + by = c \) to get \( 4x - 3y = 6 \).

Step 16: Determine Ratios for System (h)

Since the second equation \( 4x - 3y = 6 \) is rewritten in the first equation's form, all ratios \( \frac{a_1}{a_2}, \frac{b_1}{b_2}, \frac{c_1}{c_2} \) are equal, so the equations are dependent.

Key Concepts

Consistent System

A consistent system of linear equations means that there is at least one solution. In simpler terms, the lines representing the equations intersect at some point(s) on a graph. This means they either intersect at just one point, giving a unique solution, or they coincide, meaning they are the same line and thus have infinitely many solutions.

When evaluating a system to determine if it is consistent, you compare the ratios of the coefficients of the equations. For a system of equations like this:

• Equation 1: \( a_1 x + b_1 y = c_1 \)
• Equation 2: \( a_2 x + b_2 y = c_2 \)

Check these ratios:

• \( \frac{a_1}{a_2} \) and \( \frac{b_1}{b_2} \)

If the ratios of the coefficients of \( x \) and \( y \) are not equal \( (\frac{a_1}{a_2} eq \frac{b_1}{b_2}) \), then the system is consistent and has one unique solution. That's because the lines aren't parallel and will intersect at one point.

Consistency ensures that the mathematical description we've set up corresponds to some sensible situation or solves the problem we're interested in.

Inconsistent System

An inconsistent system of linear equations is where the lines do not meet at any point on a graph. This implies there is no solution to the system. When graphed, the lines are parallel and never intersect. This happens when they have the same slope but different intercepts.

To determine if a system is inconsistent, assess these equations:

• Equation 1: \( a_1 x + b_1 y = c_1 \)
• Equation 2: \( a_2 x + b_2 y = c_2 \)

You need to check:

• \( \frac{a_1}{a_2} = \frac{b_1}{b_2} \)
• Ensure \( \frac{c_1}{c_2} eq \frac{a_1}{a_2} \)

This tells you the lines are parallel but offset from each other by different intercepts. Knowing a system is inconsistent is crucial as it signals that there might be an error in your model, or the conditions being examined can't simultaneously be satisfied.

Dependent Equations

Dependent equations represent a scenario where the system of linear equations essentially describes the same line. This means the equations overlap, and thus, there are infinitely many solutions since any point on the line satisfies both equations simultaneously.

To identify dependent equations, evaluate the following system:

• Equation 1: \( a_1 x + b_1 y = c_1 \)
• Equation 2: \( a_2 x + b_2 y = c_2 \)

You confirm dependency by checking:

• \( \frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2} \)

Here, the ratios of all corresponding coefficients and constant terms are equal. This indicates that the two equations describe the same line, so every solution to one equation is also a solution to the other. Recognizing dependent equations is essential for understanding that sometimes, different forms mathematically describe the same relationship.