Last Updated on March 22, 2025 by Maged kamel
Back substitution practice problems.
First practice problem #25.
We will introduce practice problems for back substitution from Stewart’s book College Algebra: Two Practice Problems, number 25&27. For both practice problems, there are two requirements: a and b.
1-For the first practice problem, #25 for back substitution.
A matrix is given in the row-echelon form. a) write the system of equations for which the given matrix is the augmented matrix. b) use back substitution to solve the system. The following slide image shows the details of the matrix given for practice problem #25.
For part a of the first practice problem #25, we will introduce a vertical line between the third and fourth columns to show the augmented matrix.
The next step is to write the system of linear equations; we have three rows, which means that we will write three systems of equations that Can be written as follows: A—assume that we have a vertical column with three rows and one column that is (X Y Z).
The first row (1 -2 4) will be multiplied by the column vector (X Y Z). The first system of linear equations is
X-2y+4Z=3.
The second row (0 1 2) will be multiplied by the column vector (X Y Z). The second system of linear equation is 0X+Y+2Z=7.
The third row (0 0 1) is to be multiplied by the column vector (X Y Z). The third system of linear equations is 0X+0Y+1Z=2.
The next slide image shows the triangular form for the three systems of equations.
We want to get the values of X, Y, and Z. We start with the last row, for which we have a Z value equal to 2.
We move back to the second system of linear equations and substitute the value of Z in that equation to get the value of Y, equal to 7-4=3.
Check whether the estimated y is correct for problem -25.
We move back to the first system of linear equations and substitute the value of Y&Z in that equation to get the value of X equal to 3-8+6=1.
Second practice problem -27.
1-For the second practice problem #27 for back substitution. A matrix is given in the row-echelon form.
a) write the system of equations for which the given matrix is the augmented matrix. b) use back substitution to solve the system. The following slide image shows the details of the matrix given for practice problem #27.
We will introduce a vertical line between the fourth and fifth columns to show the augmented matrix.
The next step is to write the system of linear equations; we have four rows, which means that we will write four systems of equations that Can be written as follows: A—assume that we have a vertical column with three rows and one column that is (X Y Z W).
The first row (11 2 3 -1) is to be multiplied by the column vector (X Y Z W). The first system of linear equations is
X+2Y+3Z-W=7.
The second row (0 1 -2 0) is to be multiplied by the column vector (X Y Z W). The second system of linear equation is 0X+Y-2Z+0W=5.
The third row (0 0 1 2) is to be multiplied by the column vector (X Y Z W). The third system of linear equation is 0X+0Y+1Z+2W=5.
The fourth row (0 0 0 1) is to be multiplied by the column vector (X Y Z W). The fourth system of linear equation is 0X+0Y+0Z+1W=3. Two systems of linear equations are shown in the next slide image.
To get the values of X, Y, Z, and W, we start with the last row, for which we have a W value equal to 3.
We proceed to the third system of linear equations and substitute the value of W in that equation to get the value of Z equal to 5-6=-1. These are the arrangements for the 3×4 matrix to have a row echelon form.
We proceed to the second system of linear equations and substitute the values of Z&W in that equation to get the value of Y equal to 3.
Finally, we go to the first linear equation system and substitute the Y, Z, and W values in that equation to get the X value equal to 7.
Check whether the estimated x, y, and z are correct for problem -27.
It is essential to check the Values of X, Y, and Z in any of the four systems of equations to ensure that the solution is accurate. The next slide shows the steps I use to check the x,y,z, and w values for all the given equations.
In the next post, we will solve practice problems for Gauss Jordan elimination.
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For a valid external link, math is fun for the matrix part.