Problem 13 Solve each system using Gaussian... [FREE SOLUTION]

Chapter 9: Problem 13

Solve each system using Gaussian elimination. $$\begin{aligned}x+4 y &=-1 \\\3 x+5 y &=4\end{aligned}$$

Short Answer

Expert verified

The solution to the given system of linear equations using Gaussian elimination is $x = 3$ and $y = -1$.

Step by step solution

Write the system in the form of an augmented matrix

First, convert the given system of linear equations into an augmented matrix: $$ \left[ \begin{array}{cc|c} 1 & 4 & -1 \\ 3 & 5 & 4 \end{array} \right] $$

Use row operations to convert the matrix into the row-echelon form

To achieve row-echelon form, we'll use row operations. Starting by eliminating '3' in the second row and the first column, we'll perform the following row operation: Row2 = Row2 - (3 * Row1) $$ \left[ \begin{array}{cc|c} 1 & 4 & -1 \\ 0 & -7 & 7 \end{array} \right] $$ Now the matrix is in row-echelon form.

Convert the row-echelon form back into the equivalent system of linear equations

Now, we can convert the row-echelon form back into the equivalent system of linear equations: $$ \begin{aligned} x+4 y &=-1 \\ -7y &= 7 \end{aligned} $$

Solve the simplified system for x and y

Now we can solve for y using the second equation: Divide both sides of the equation by -7 to obtain the value of y: $y = \frac{7}{7} = -1$ Now that we have the value for y, we can substitute it back into the first equation to solve for x: $x + 4(-1) = x-4 = -1$ Adding 4 to both sides of the equation gives: $x = 3$ So, the solution to the given system of linear equations using Gaussian elimination is: $x = 3$ and $y = -1$.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Linear Equations

Linear equations are mathematical expressions that represent a relationship between variables where the variables have a constant exponent of 1. In simpler terms, they are equations that plot straight lines when graphed on a coordinate plane.
In the exercise example, we have a system of two linear equations:

Equation 1: $ x + 4y = -1 $
Equation 2: $ 3x + 5y = 4 $

These linear equations represent straight lines that intersect at a particular point, which provides the solution for $x$ and $y$. Solving systems of linear equations involves finding values for variables that satisfy all equations simultaneously. This often requires techniques such as substitution, elimination, or matrix methods like Gaussian elimination.

Augmented Matrix

An augmented matrix is a compact way of representing a system of linear equations. It includes not only the coefficients of the variables but also the solutions (constants) on the right-hand side.
The augmented matrix transforms the system:

Equation 1: $ x + 4y = -1 $
Equation 2: $ 3x + 5y = 4 $

into the following matrix:\[\begin{array}{cc|c}1 & 4 & -1 \3 & 5 & 4\end{array}\]The columns before the vertical line represent coefficients of $x$ and $y$. The column after the line represents the constants from the linear equations. Thus, the augmented matrix is a great way to visualize and perform operations for solving the system.

Row Operations

Row operations are techniques used to manipulate matrices to simplify solving systems of equations. There are three main types of row operations:

Row swapping: interchange two rows of the matrix.
Row multiplication: multiply all elements of a row by a non-zero scalar.
Row addition: add or subtract a multiple of one row to another row.

In our exercise, we used row addition to help simplify the augmented matrix. Specifically, to zero out the coefficient of $x$ in the second row, we performed the operation:
Row2 = Row2 - 3 * Row1:
\[\begin{array}{cc|c}1 & 4 & -1 \0 & -7 & 7\end{array}\]This operation maintains the equivalence of the system while making the matrix easier to work with for solving the unknowns.

Row-Echelon Form

Row-echelon form is a specific form of a matrix achieved through row operations. This form helps to easily determine the solutions of a system of linear equations.
In row-echelon form, the following conditions are typically met:

All nonzero rows are above any rows of all zeroes.
The leading coefficient (first non-zero number from the left, also called the pivot) of a non-zero row is always to the right of the leading coefficient of the row above it.
It's common for the leading coefficient to be 1.

In our example, the system was transformed into:\[\begin{array}{cc|c}1 & 4 & -1 \0 & -7 & 7\end{array}\]This matrix meets the conditions to be considered in row-echelon form and allows us to then easily backtrack to solve for $y$ and $x$ using back-substitution.

91影视

Solve each system using Gaussian elimination. $$\begin{aligned}x+4 y &=-1 \\\3 x+5 y &=4\end{aligned}$$

Short Answer

Step by step solution

Write the system in the form of an augmented matrix

Use row operations to convert the matrix into the row-echelon form

Convert the row-echelon form back into the equivalent system of linear equations

Solve the simplified system for x and y

Key Concepts

Linear Equations

Augmented Matrix

Row Operations

Row-Echelon Form

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