Problem 11 - Calculus for Biology and Medicine

Question

Reduce the system of linear equations to upper triangular form and solve. $$ \begin{array}{l} 7 x-y=4 \\ 3 x+2 y=1 \end{array} $$

Step-by-Step Solution

Answer

The solution is \( x = \frac{9}{17}, y = -\frac{5}{17} \).

Step 1: Understanding the System of Equations

We have a system of two linear equations: \( 7x - y = 4 \) and \( 3x + 2y = 1 \). The goal is to convert this system into upper triangular form and solve for \(x\) and \(y\).

Step 2: Forming the Augmented Matrix

Write the augmented matrix for the system of equations: \[ \begin{bmatrix} 7 & -1 & | & 4 \ 3 & 2 & | & 1 \end{bmatrix}\]. This matrix represents the coefficients of \(x\) and \(y\) along with the constants from both equations.

Step 3: Eliminating \(x\) in the Second Row

To form an upper triangular matrix, eliminate the \(x\) term from the second equation. Multiply the first row by 3 and the second row by 7:\[\begin{bmatrix} 21 & -3 & | & 12 \ 21 & 14 & | & 7\end{bmatrix}\]. Now subtract the first row from the second row.

Step 4: Subtract Rows to Eliminate \(x\)

Subtracting the first row from the second row (\(R_2 = R_2 - R_1\)) gives:\[\begin{bmatrix} 7 & -1 & | & 4 \ 0 & 17 & | & -5\end{bmatrix}\]. This matrix is now in upper triangular form.

Step 5: Solving for \(y\)

Now that we have 0s under the first pivot, we can solve for \(y\) from the second equation: \[ 17y = -5 \]. Solve for \(y\) to get \( y = -\frac{5}{17} \).

Step 6: Back Substitution to Solve for \(x\)

With the value of \(y\), substitute back into the first equation: \[ 7x - \left(-\frac{5}{17}\right) = 4 \]. This simplifies to \(7x + \frac{5}{17} = 4\). Multiply the entire equation by 17 to clear the fraction: \[ 119x + 5 = 68 \]. Solve for \(x\) to get \( x = \frac{63}{119} = \frac{9}{17} \).

Key Concepts

Linear Equations

Linear equations are expressions that represent a line when graphed on a coordinate plane. They typically take the form of \( ax + by = c \), where \( a \), \( b \), and \( c \) are constants, and \( x \) and \( y \) are variables. In the context of a system of linear equations, we are dealing with multiple such equations that share common variables. The goal is to find values of the variables that satisfy all equations in the system. To solve these systems, we can use various methods such as substitution, elimination, or matrix techniques. Each method has its own advantages and is chosen based on the complexity of the system.

Augmented Matrix

An augmented matrix is a helpful tool for solving a system of linear equations. It is constructed by aligning the coefficients of variables and the constants from each equation into a matrix format. For the given system:

• \( 7x - y = 4 \)
• \( 3x + 2y = 1 \)

The augmented matrix is written as:\[\begin{bmatrix}7 & -1 & | & 4 \3 & 2 & | & 1\end{bmatrix}\]Each row of the matrix represents an equation, whereas the vertical line denotes the separation between the coefficients and the constants. The advantage of using an augmented matrix is that it facilitates systematic manipulation of the equations, particularly when using the elimination method to achieve upper triangular form, which makes solving the system more straightforward.

Back Substitution

Back substitution is a systematic method used to solve variables in a system of linear equations that has been reduced to upper triangular form. Once the matrix falls into upper triangular form:\[\begin{bmatrix}7 & -1 & | & 4 \0 & 17 & | & -5\end{bmatrix}\]The idea is to solve the equations from bottom to top, starting with the equation where the variable appears alone. In our system, we start with the second equation \(17y = -5\) and solve for \(y\), giving \( y = -\frac{5}{17} \).
With \(y\) known, back substitution involves replacing \(y\) in the first equation to solve for \(x\). The first equation is manipulated into a form that isolates \(x\):
\[7x + \frac{5}{17} = 4\]
Multiplying through by 17 clears the fraction, making it straightforward to find \(x\). The process of back substitution efficiently narrows down to exact solutions without guesswork.