Lesson OVERVIEW of LESSONS on determinants of 3x3-matrices and Cramer's rule for systems in 3 unknowns

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<H2>OVERVIEW of LESSONS on determinants of 3x3-matrices and Cramer's rule for systems in 3 unknowns</H2>

For your convenience this file contains the list of my lessons on determinants of 3x3 matrices and the Cramer's rule for systems in 3 unknowns in the logical order.

Then the lessons are listed again with short annotations of their contents.

<H3>The list of lessons</H3>&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Determinant-of-a-3x3-matrix.lesson>Determinant of a 3x3 matrix</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Co-factoring-a-3x3-determinant.lesson>Co-factoring the determinant of a 3x3 matrix</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/HOW-TO-solve-system-of-linear-eqns-in-three-unknowns-using-det.lesson>HOW TO solve system of linear equations in three unknowns using determinant (Cramer's rule)</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-linear-equations-in-three-unknowns-using-determinant.lesson>Solving systems of linear equations in three unknowns using determinant (Cramer's rule)</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-word-problems-by-reducing-them-to-systems-of-linear-equations-in-three-unknowns.lesson>Solving word problems by reducing to systems of linear equations in three unknowns</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/The-trick-to-solve-some-word-problems-with-three-and-more-unknowns.lesson>The tricks to solve some word problems with three and more unknowns using mental math</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-non-linear-equations-in-three-unknowns-using-Cramer%27s-rule.lesson>Solving systems of non-linear equations in three unknowns using Cramer's rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Sometime-two-eqns-are-enough-to-find-three-unknowns-by-an-UNIQUE-way.lesson>Sometime two equations are enough to find three unknowns by an UNIQUE way</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Two-very-different-approaches-to-one-word-problem.lesson>Two very different approaches to one word problem</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-word-problems-in-three-unknowns-by-the-backward-method.lesson>Solving word problems in three unknowns by the backward method</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-a-system-of-linear-equations-in-17-unknowns.lesson>Solving system of linear equation in 17 unknowns</A>

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-a-system-of-linear-equations-in-19-unknowns.lesson>Solving system of linear equation in 19 unknowns</A>

<H3>The lessons with short annotations</H3>Lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Determinant-of-a-3x3-matrix.lesson>Determinant of a 3x3 matrix</A>

<TABLE> 
 <TR>
 <TD>&nbsp;&nbsp;&nbsp;&nbsp;The &nbsp;determinant&nbsp; of the matrix &nbsp;{{{A}}}&nbsp; = 
 </TD>
 <TD>&nbsp;{{{drawing( 101, 75, -0.5, 9.6, 0, 7.5, 
 locate (0.4, 8.3, a[11]),
 locate (3.4, 8.3, a[12]),
 locate (6.6, 8.3, a[13]),
 locate (0.4, 5.9, a[21]),
 locate (3.4, 5.9, a[22]),
 locate (6.6, 5.9, a[23]),
 locate (0.4, 3.3, a[31]),
 locate (3.4, 3.3, a[32]),
 locate (6.6, 3.3, a[33]),

arc (-1.0, 3.25, 22.0, 22.0, 330, 30),
            arc (10.4, 3.25, 22.0, 22.0, 150, 210)
)}}}

</TD>
 <TD>&nbsp;is the number &nbsp;&nbsp;{{{d}}} = {{{a[11]a[22]a[33]}}} + {{{a[12]a[23]a[31]}}} + {{{a[13]a[21]a[32]}}} - {{{a[13]a[22]a[31]}}} - {{{a[12]a[21]a[33]}}} - {{{a[11]a[32]a[23]}}}. 
 </TD>
 </TR>
</TABLE><TABLE> 
 <TR>
 <TD>
&nbsp;&nbsp;&nbsp;&nbsp;Thus the determinant is defined for any square matrix with &nbsp;3&nbsp; rows and &nbsp;3&nbsp; columns.<TABLE> 
 <TR>
 <TD>&nbsp;&nbsp;&nbsp;&nbsp;Sometimes the determinant of a matrix
 </TD>
 <TD>&nbsp;&nbsp;{{{drawing( 101, 75, -0.5, 9.6, 0, 7.5, 
 locate (0.4, 8.3, a[11]),
 locate (3.4, 8.3, a[12]),
 locate (6.6, 8.3, a[13]),
 locate (0.4, 5.9, a[21]),
 locate (3.4, 5.9, a[22]),
 locate (6.6, 5.9, a[23]),
 locate (0.4, 3.3, a[31]),
 locate (3.4, 3.3, a[32]),
 locate (6.6, 3.3, a[33]),

arc (-1.0, 3.25, 22.0, 22.0, 330, 30),
 arc (10.4, 3.25, 22.0, 22.0, 150, 210)
)}}}
 </TD>
 <TD>&nbsp;is denoted as &nbsp;det
 </TD>
 <TD>{{{drawing( 101, 75, -0.5, 9.6, 0, 7.5, 
 locate (0.4, 8.3, a[11]),
 locate (3.4, 8.3, a[12]),
 locate (6.6, 8.3, a[13]),
 locate (0.4, 5.9, a[21]),
 locate (3.4, 5.9, a[22]),
 locate (6.6, 5.9, a[23]),
 locate (0.4, 3.3, a[31]),
 locate (3.4, 3.3, a[32]),
 locate (6.6, 3.3, a[33]),

arc (-1.0, 3.25, 22.0, 22.0, 330, 30),
 arc (10.4, 3.25, 22.0, 22.0, 150, 210)
)}}}&nbsp;&nbsp;&nbsp;or
 </TD>
 <TD>&nbsp;&nbsp;&nbsp;{{{drawing( 96, 75, 0.0, 9.6, 0, 7.5, 
 locate (0.4, 8.3, a[11]),
 locate (3.4, 8.3, a[12]),
 locate (6.6, 8.3, a[13]),
 locate (0.4, 5.9, a[21]),
 locate (3.4, 5.9, a[22]),
 locate (6.6, 5.9, a[23]),
 locate (0.4, 3.3, a[31]),
 locate (3.4, 3.3, a[32]),
 locate (6.6, 3.3, a[33]),

line (0.0, 7.5, 0.0, 0.0),
 line (9.5, 7.5, 9.5, 0.0)
)}}}.
 </TD>
 </TR>
</TABLE>
<TABLE> 
 <TR>
 <TD>&nbsp;&nbsp;&nbsp;&nbsp;So, &nbsp;&nbsp;det
 </TD>
 <TD>{{{drawing( 101, 75, -0.5, 9.6, 0, 7.5, 
 locate (0.4, 8.3, a[11]),
 locate (3.4, 8.3, a[12]),
 locate (6.6, 8.3, a[13]),
 locate (0.4, 5.9, a[21]),
 locate (3.4, 5.9, a[22]),
 locate (6.6, 5.9, a[23]),
 locate (0.4, 3.3, a[31]),
 locate (3.4, 3.3, a[32]),
 locate (6.6, 3.3, a[33]),

arc (-1.0, 3.25, 22.0, 22.0, 330, 30),
            arc (10.4, 3.25, 22.0, 22.0, 150, 210)
)}}}

</TD>
 <TD> = 
 </TD>
 <TD>&nbsp;&nbsp;{{{drawing( 96, 75, 0.0, 9.6, 0, 7.5, 
 locate (0.4, 8.3, a[11]),
 locate (3.4, 8.3, a[12]),
 locate (6.6, 8.3, a[13]),
 locate (0.4, 5.9, a[21]),
 locate (3.4, 5.9, a[22]),
 locate (6.6, 5.9, a[23]),
 locate (0.4, 3.3, a[31]),
 locate (3.4, 3.3, a[32]),
 locate (6.6, 3.3, a[33]),

line (0.0, 7.5, 0.0, 0.0),
            line (9.5, 7.5, 9.5, 0.0)
)}}}

</TD>
 <TD> = {{{a[11]a[22]a[33]}}} + {{{a[12]a[23]a[31]}}} + {{{a[13]a[21]a[32]}}} - {{{a[13]a[22]a[31]}}} - {{{a[12]a[21]a[33]}}} - {{{a[11]a[32]a[23]}}}. 
 </TD>
 </TR>
</TABLE><TABLE> 
 <TR>
 <TD>
&nbsp;&nbsp;&nbsp;&nbsp;The patterns for calculation the determinant terms are shown in &nbsp;Figure 1a&nbsp; and &nbsp;1b.

&nbsp;&nbsp;&nbsp;&nbsp;Each term of a determinant, &nbsp;additive or subtrahend, &nbsp;is the product of elements that&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;are located in different rows and different columns of a matrix.

&nbsp;&nbsp;&nbsp;&nbsp;No one single term of a determinant contains two elements from the same row. &nbsp;No one

&nbsp;&nbsp;&nbsp;&nbsp;single term of a determinant contains two elements from the same column.

</TD>
 <TD>
&nbsp;&nbsp;&nbsp;&nbsp;{{{drawing( 96, 75, 0.0, 9.6, 0, 7.5, 
 circle (1.5, 6.4, 0.2),
 circle (4.7, 6.4, 0.2),
 circle (8.0, 6.4, 0.2),
 circle (1.5, 3.8, 0.2),
 circle (4.7, 3.8, 0.2),
 circle (8.0, 3.8, 0.2),
 circle (1.6, 1.2, 0.2),
 circle (4.7, 1.2, 0.2),
 circle (8.0, 1.2, 0.2),

line (0.0, 7.5, 0.0, 0.0),
            line (9.5, 7.5, 9.5, 0.0),

line (1.5, 6.4, 4.7, 3.8), 
            line (4.7, 3.8, 8.0, 1.2),

line (8.0, 6.4, 1.5, 3.8), 
            line (8.0, 6.4, 4.7, 1.2), 
            line (1.5, 3.8, 4.7, 1.2),

line (1.6, 1.2, 4.7, 6.4), 
            line (1.6, 1.2, 8.0, 3.8), 
            line (4.7, 6.4, 8.0, 3.8)
)}}}

Fig. 1a. (+)-Patterns&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp; 
for the determinant

line (0.0, 7.5, 0.0, 0.0),
            line (9.5, 7.5, 9.5, 0.0),

red(line (1.6, 1.2, 4.7, 3.8)),
        red(line (4.7, 3.8, 8.0, 6.4)),

red(line (1.5, 6.4, 4.7, 1.2)),
        red(line (1.5, 6.4, 8.0, 3.8)),
        red(line (4.7, 1.2, 8.0, 3.8)),

red(line (1.5, 3.8, 4.7, 6.4)),
        red(line (1.5, 3.8, 8.0, 1.2)),
        red(line (4.7, 6.4, 8.0, 1.2))
)}}}

Fig. 1b. (-)-Patterns 
for the determinant
 </TD>
 </TR>
</TABLE>

&nbsp;&nbsp;&nbsp;&nbsp;Examples. &nbsp;Calculate determinants of matrices

&nbsp;&nbsp;&nbsp;&nbsp;{{{(matrix(3,3, 0,0,0,0,0,0,0,0,0))}}}, &nbsp;&nbsp;&nbsp;&nbsp;{{{(matrix(3,3, 1,0,0,0,1,0,0,0,1))}}}, &nbsp;&nbsp;&nbsp;&nbsp;{{{(matrix(3,3, 1,0,0,0,2,0,0,0,3))}}}, &nbsp;&nbsp;&nbsp;&nbsp;{{{(matrix(3,3, a,0,0,0,b,0,0,0,c))}}}, &nbsp;&nbsp;&nbsp;&nbsp;{{{(matrix(3,3, 1,2,3,0,4,5,0,0,6))}}}, &nbsp;&nbsp;&nbsp;&nbsp;{{{(matrix(3,3, a,x,y,0,b,z,0,0,c))}}}, &nbsp;&nbsp;&nbsp;&nbsp;{{{(matrix(3,3, 1,0,0,2,3,0,4,5,6))}}}, &nbsp;&nbsp;&nbsp;&nbsp;{{{(matrix(3,3, a,0,0,x,b,0,y,z,c))}}}, &nbsp;&nbsp;&nbsp;&nbsp;{{{(matrix(3,3, 1,-3,5,7,2,-3,5,-1,-2))}}}, &nbsp;&nbsp;&nbsp;&nbsp;{{{(matrix(3,3, 0,0,0,7,2,-3,5,-1,-2))}}}, &nbsp;&nbsp;&nbsp;&nbsp;{{{(matrix(3,3, 1,0,0,7,2,-3,5,-1,-2))}}}, &nbsp;&nbsp;&nbsp;&nbsp;{{{(matrix(3,3, 0,0,5,7,2,-3,5,-1,-2))}}}.

Lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/HOW-TO-solve-system-of-linear-eqns-in-three-unknowns-using-det.lesson>HOW TO solve system of linear equations in three unknowns using determinant (Cramer's rule)</A>

&nbsp;&nbsp;&nbsp;&nbsp;If

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(a*x + b*y + c*z = u, 
d*x + e*y + f*z = v, 
g*x + h*y + k*z = w)}}}

&nbsp;&nbsp;&nbsp;&nbsp;is a system of three linear equations in three unknown &nbsp;{{{x}}}, &nbsp;{{{y}}}&nbsp; and &nbsp;{{{z}}}, &nbsp;where &nbsp;{{{a}}}, &nbsp;{{{b}}}, &nbsp;{{{c}}}, &nbsp;{{{d}}}, &nbsp;{{{e}}}, &nbsp;{{{f}}}, &nbsp;{{{g}}}, &nbsp;{{{h}}} &nbsp;and&nbsp; {{{k}}}&nbsp; are coefficients and &nbsp;{{{u}}}, &nbsp;{{{v}}}&nbsp; and &nbsp;{{{w}}}&nbsp; are right side

&nbsp;&nbsp;&nbsp;&nbsp;constants, &nbsp;then the solution of this system is &nbsp;(the Cramer's rule)

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{x = det(matrix(3,3,u,b,c,v,e,f,w,h,k))/det(matrix(3,3,a,b,c,d,e,f,g,h,k))}}}, &nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{y = det(matrix(3,3,a,u,c,d,v,f,g,w,k))/det(matrix(3,3,a,b,c,d,e,f,g,h,k))}}},&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{z = det(matrix(3,3,a,b,u,d,e,v,g,h,w))/det(matrix(3,3,a,b,c,d,e,f,g,h,k))}}}.&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;

&nbsp;&nbsp;&nbsp;&nbsp;Here the matrix &nbsp;{{{(matrix(3,3,a,b,c,d,e,f,g,h,k))}}}&nbsp; in the denominators is the coefficient matrix of the system formed by its coefficients. &nbsp;It is assumed that the coefficient matrix

&nbsp;&nbsp;&nbsp;&nbsp;has a non-zero determinant. &nbsp;This is the necessary  condition for the Cramer's rule applicability.

&nbsp;&nbsp;&nbsp;&nbsp;The matrix in the numerator for the &nbsp;first unknown&nbsp; {{{x}}} &nbsp;is obtained from the coefficient matrix after replacing its &nbsp;first column&nbsp; by the right sides vector &nbsp;{{{(matrix(3, 1, u, v, w))}}}.

&nbsp;&nbsp;&nbsp;&nbsp;The matrix in the numerator for the &nbsp;second unknown&nbsp; {{{y}}} &nbsp;is obtained from the coefficient matrix after replacing its &nbsp;second column&nbsp; by the right sides vector &nbsp;{{{(matrix(3,1, u, v, w))}}}.

&nbsp;&nbsp;&nbsp;&nbsp;The matrix in the numerator for the &nbsp;third unknown&nbsp; {{{z}}} &nbsp;is obtained from the coefficient matrix after replacing its &nbsp;third column&nbsp; by the right sides vector &nbsp;{{{(matrix(3,1, u, v, w))}}}.

&nbsp;&nbsp;&nbsp;&nbsp;Examples 1&nbsp; and &nbsp;2. &nbsp;Solve theses systems of three linear equations in three unknowns

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(x + 2y - 3z = -4,
x + y + z = 6,
2x - y + z = 3)}}}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(y + 2z = -1,
x + y = -1,
2x - z = 3)}}}

Lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-linear-equations-in-three-unknowns-using-determinant.lesson>Solving systems of linear equations in three unknowns using determinant (Cramer's rule)</A>

&nbsp;&nbsp;&nbsp;&nbsp;Problems 1, &nbsp;2&nbsp; and &nbsp;3. &nbsp;Solve these systems of three linear equations in three unknowns

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(2x + 2y - z = 2,
x + y + z = 4,
x - y - z = 2)}}}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(y - 2z = -3,
x - y = 2,
3x + z = 11)}}}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(x + y + z = 0,
x - y - 3z = 0,
x + y - 2z = 0)}}}

&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;Solve the system of three non-linear equations in three unknowns

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(1/x + 1/y - 1/z = 4, 
-1/x + 1/y + 1/z = 6,
1/x + 1/y + 1/z = 8)}}}

Lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-word-problems-by-reducing-them-to-systems-of-linear-equations-in-three-unknowns.lesson>Solving word problems by reducing to systems of linear equations in three unknowns</A>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;A concert audience of &nbsp;400&nbsp; people consists of adults, &nbsp;students, &nbsp;and children. &nbsp;The ticket prices are &nbsp;$40&nbsp; for adults, &nbsp;$20&nbsp; for students, &nbsp;and &nbsp;$10&nbsp; for

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;children. &nbsp;The total amount of money taken in is &nbsp;$10600. &nbsp;The number of children tickets sold is in &nbsp;200&nbsp; less than the number of adult and student tickets

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;in total. &nbsp;How many adults, &nbsp;students, &nbsp;and children are in attendance?

&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Maria invested a total &nbsp;$26000&nbsp; into three accounts: a saving account paying &nbsp;3%&nbsp; interest, &nbsp;a &nbsp;CD&nbsp; deposit paying &nbsp;4%&nbsp; interest, &nbsp;and a bond paying &nbsp;5%&nbsp; interest.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Her total annual interest from the three investments was &nbsp;$1000. &nbsp;The interest from the saving account was &nbsp;$460&nbsp; less than the total interest from the other

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;two investments. &nbsp;How much did she invest at each rate?

&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;Ron attends a cocktail party &nbsp;(with his graphing calculator in his pocket). &nbsp;He wants to limit his food intake to

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;109 g &nbsp;protein, &nbsp;103 g &nbsp;fat, &nbsp;and &nbsp;153 g &nbsp;carbohydrate.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;According to the health conscious hostess, &nbsp;the marinated mushroom caps have &nbsp;3 g &nbsp;protein, &nbsp;5 g &nbsp;fat, &nbsp;and &nbsp;9 g &nbsp;carbohydrate;

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;the spicy meatballs have &nbsp;14 g &nbsp;protein, &nbsp;7 g &nbsp;fat, &nbsp;and &nbsp;15 g &nbsp;carbohydrate;

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and the deviled eggs have &nbsp;13 g &nbsp;protein, &nbsp;15 g &nbsp;fat, &nbsp;and &nbsp;6 g &nbsp;carbohydrate.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;How many of each snack can he eat to obtain his goal&nbsp;?

&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;There were 240 persons in a picnic. There were 20 more men than women and 20 more adults than children.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;How many men were there in the picnic ?

Lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/The-trick-to-solve-some-word-problems-with-three-and-more-unknowns.lesson>The tricks to solve some word problems with three and more unknowns using mental math</A>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Alan, &nbsp;Bob and Cleve weight together &nbsp;255 pounds. &nbsp;Bob, &nbsp;Cleve and Dan weight together &nbsp;267 pounds.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Cleve, &nbsp;Dan and Alan weight together &nbsp;262 pounds. &nbsp;Dan, &nbsp;Alan and Bob weight together &nbsp;257 pounds.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;What is the weight of each of the boys?

&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;Jonny has a collection of coins in nickels &nbsp;(5 cents), &nbsp;dimes &nbsp;(10 cents) and the quarters &nbsp;(25 cents).

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The amount of money in nickels and dimes in the collection is of &nbsp;130&nbsp; cents.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The amount of money in nickels and quarters is of &nbsp;285&nbsp; cents. &nbsp;The amount of money in dimes and quarters is of &nbsp;295&nbsp; cents.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;How many coins of each nomination are there in the collection?

&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;Alice and Barbara weight together in &nbsp;50&nbsp; pounds more than Christi weights.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Alice and Christi weight together in &nbsp;52&nbsp; pounds more than Barbara weights.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Barbara and Christi weight together in &nbsp;54&nbsp; pounds more than Alice weights.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Find the weight of each of the girls.

&nbsp;&nbsp;&nbsp;&nbsp;Problem 4. &nbsp;Five baskets contain coconuts. &nbsp;The first and second baskets together have a total of &nbsp;52 coconuts.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The second and third baskets have &nbsp;43 coconuts. &nbsp;The third and fourth baskets have &nbsp;34 coconuts.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The fourth and fifth baskets have &nbsp;30 coconuts, &nbsp;and the first and fifth baskets have &nbsp;47 coconuts.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;How many coconuts are in each basket?

&nbsp;&nbsp;&nbsp;&nbsp;Problem 5. &nbsp;In a triangle, &nbsp;the sum of the measures of its first and the second side is in &nbsp;2 cm&nbsp; greater than the measure of the third side.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The sum of the measures of its first and the third side is in &nbsp;4 cm&nbsp; greater than the measure of the second side.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The sum of the measures of its second and the third side is in &nbsp;6 cm&nbsp; greater than the measure of the first side.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Find the perimeter and the measure of each side of the triangle.

&nbsp;&nbsp;&nbsp;&nbsp;Problem 6. &nbsp;In a quadrilateral, &nbsp;the sum of the measures of its first, &nbsp;the second and the third side is &nbsp;12 cm.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The sum of the measures of its first, &nbsp;the third side and the fourth side is &nbsp;14 cm.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The sum of the measures of its first, &nbsp;the third and the fourth side is &nbsp;16 cm.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The sum of the measures of its second, &nbsp;the third and the fourth side is &nbsp;18 cm.

Lesson &nbsp;<A HREF=http://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-systems-of-non-linear-equations-in-three-unknowns-using-Cramer%27s-rule.lesson>Solving systems of non-linear equations in three unknowns using Cramer's rule</A>

&nbsp;&nbsp;&nbsp;&nbsp;Problems 1, 2 and 3. &nbsp;Solve these systems of three non-linear equations in three unknowns

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(1/x + 2/y - 2/z = -2,
5/x + 9/y - 4/z = -3,
3/x + 4/y - 5/z = -3)}}}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(3/(x+2) + 2/(y-1) - 1/(z-3) = 1,
2/(x+2) + 3/(y-1) - 1/(z-3) = 1,
1/(x+2) - 1/(y-1) + 2/(z-3) = 3)}}}&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;{{{system(3/sqrt(x-1) + 1/sqrt(y-2) - 1/sqrt(z-3) = 2,
3/sqrt(x-1) - 1/sqrt(y-2) + 2/sqrt(z-3) = 7,
-3/sqrt(x-1) + 2/sqrt(y-2) - 1/sqrt(z-3) = -2)}}}

<A HREF=https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Sometime-two-eqns-are-enough-to-find-three-unknowns-by-an-UNIQUE-way.lesson>Sometime two equations are enough to find three unknowns by an UNIQUE way</A>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;"Sue's Candies" &nbsp;has a candy stall in the market.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;One candy costs &nbsp;5 &nbsp;cents each, &nbsp;the second costs &nbsp;20 &nbsp;cents each and the &nbsp;3rd &nbsp;costs &nbsp;30 &nbsp;cents each.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;In her absence due to personal necessity an honest buyer picked a total of &nbsp;20 &nbsp;pieces of candies

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;costing &nbsp;5 &nbsp;cents, &nbsp;20 &nbsp;cents and &nbsp;30 &nbsp;cents in the stall and left &nbsp;2 &nbsp;pesos for their exact cost.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;How many each candy of &nbsp;5 &nbsp;cents, &nbsp;20 &nbsp;cents, &nbsp;and &nbsp;30 &nbsp;cents the honest buyer picked?

&nbsp;&nbsp;&nbsp;&nbsp;Problem 2. &nbsp;A farmer wants to buy 100 animals for $100.00 (must buy at least one of them)
<pre>
 Horses cost $5.00 each
 Cows cost $3.00 each
 Chickens cost $0.50 each.</pre>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;How many animals the farmer should buy ?

&nbsp;&nbsp;&nbsp;&nbsp;Problem 3. &nbsp;A furniture company makes dining room furniture.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;A buffet requires &nbsp;40 &nbsp;hours for construction 10 hours for finishing.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;A chair requires &nbsp;20 hours for construction and &nbsp;20 hours for finishing.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;A table requires &nbsp;40 hours for construction and &nbsp;70 hours for finishing.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;The construction department has &nbsp;500 hours of labor

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;and the finishing department has &nbsp;200 hours of labor available each week.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;How many pieces of each type of furniture should be produced each week if the factory is to run at full capacity?

<A HREF=https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Two-very-different-approaches-to-one-word-problem.lesson>Two very different approaches to one word problem</A>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Four friends, &nbsp;Aisha, &nbsp;Ben, &nbsp;Cathy, &nbsp;and &nbsp;Daniel shared lunch equally.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Aisha paid &nbsp;3/5 &nbsp;of the total amount &nbsp;Ben and &nbsp;Cathy paid.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Cathy paid &nbsp;$10 &nbsp;more than &nbsp;Ben.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;Daniel repaid &nbsp;$24.00 &nbsp;to &nbsp;Aisha and some money to &nbsp;Ben and &nbsp;Cathy.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;How much did &nbsp;Daniel repaid to Ben and how much to &nbsp;Cathy?

<A HREF=https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-word-problems-in-three-unknowns-by-the-backward-method.lesson>Solving word problems in three unknowns by the backward method</A>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;Alice, &nbsp;Ben and &nbsp;Carl collect stamps. &nbsp;They exchange stamps among themselves according to the following scheme:

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(1) &nbsp;&nbsp;Alice gives &nbsp;Ben as many stamps as &nbsp;Ben has and &nbsp;Carl as many stamps as &nbsp;Carl has.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(2) &nbsp;&nbsp;After that, &nbsp;Ben gives &nbsp;Alice and &nbsp;Carl as many stamps as each of them has,

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;(3) &nbsp;&nbsp;and then &nbsp;Carl gives &nbsp;Alice and &nbsp;Ben as many stamps as each has.

&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;If each finally has &nbsp;80 &nbsp;stamps, &nbsp;with how many stamps does &nbsp;Ben start?

<A HREF=https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-a-system-of-linear-equations-in-17-unknowns.lesson>Solving system of linear equation in 17 unknowns</A>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If &nbsp;({{{a[1]}}}, {{{a[2]}}}, . . . , {{{a[17]}}}) &nbsp;satisfy
<pre>
 {{{a[1] + a[2] + a[3]}}} = 1, 
 {{{a[2] + a[3] + a[4]}}} = 2,
 {{{a[3] + a[4] + a[5]}}} = 3, 
 . . . . . . . . . 
 {{{a[15] + a[16] + a[17]}}} = 15, 
 {{{a[16] + a[17] + a[1]}}} = 16, 
 {{{a[17] + a[1] + a[2]}}} = 17,
</pre>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;find the value of &nbsp;{{{a[17]}}}.

<A HREF=https://www.algebra.com/algebra/homework/Matrices-and-determiminant/Solving-a-system-of-linear-equations-in-19-unknowns.lesson>Solving system of linear equation in 19 unknowns</A>

&nbsp;&nbsp;&nbsp;&nbsp;Problem 1. &nbsp;If &nbsp;{{{a[1]}}}, {{{a[2]}}}, . . ., {{{a[19]}}} &nbsp;satisfy
<pre>
 {{{a[1] + a[2] + a[3] + a[4]}}} = 1, 
 {{{a[2] + a[3] + a[4] + a[5]}}} = 2, 
 {{{a[3] + a[4] + a[5] + a[6]}}} = 3, 
 . . . . . . . . . 
 {{{a[16] + a[17] + a[18] + a[19]}}} = 16, 
 {{{a[17] + a[18] + a[19] + a[1]}}} = 17, 
 {{{a[18] + a[19] + a[1] + a[2]}}} = 18, 
 {{{a[19] + a[1] + a[2] + a[3]}}} = 19, 
</pre>&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;&nbsp;find the value of &nbsp;{{{a[19]}}}.

Use this file/link &nbsp;<A HREF=https://www.algebra.com/algebra/homework/complex/ALGEBRA-II-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-II - YOUR ONLINE TEXTBOOK</A>&nbsp; to navigate over all topics and lessons of the online textbook &nbsp;ALGEBRA-II.