ALGEBRA - solved problems
System of linear equations word problems
 107 A two-digit number enlarges by nine when its digits reverse. The same two-digit number divided by
sum of its digits gives quotient 5 and reminder 4. Find the two-digit number.
Solution:    Let x be ten's digit and y units' digit, then 10x+ y is a two-digit number.
Then, written are conditions of the given problem: 108 An isosceles triangle with the sides 6 cm longer then base has the perimeter 48 cm.
What is the length of its base and the sides?
 Solution: 109 The perimeter of a rectangle is 42 cm. The ratio between its width and the length is 3 : 4.
Find the length and width of the rectangle.
 Solution: Solving systems of equations graphically
 110 Solve graphically given system of linear equations.
Solution:

 l1 ::   2x + 3y - 4 = 0 l2 ::  -x + 2y - 5 = 0 Coefficients satisfy the condition: so, the lines intersect. Cramer’s rule (using the determinant) to solve systems of linear equations
Solving system of two equations in two unknowns using Cramer's rule
A system of two equations in two unknowns, the solution to a system by Cramer’s rule (use of determinants). the solution to the system 111 Solve given system of linear equations using Cramer’s rule.
 Solution: Solving system of three equations in three unknowns using Cramer's rule A determinant of rank n can be evaluated by expanding to its cofactors of rank n - 1, along any row or column taking into account the scheme of the signs. For example, the determinant of rank n = 3, 112 Solve given system of three equations in three unknowns using method of expanding to cofactors.
 Solution: Method of expanding a determinant of a rank n to cofactors
The value of a determinant will not change by adding multiples of any column or row to any other column or row. This way created are zero entries that simplify subsequent calculations.
 113 Solve given determinant of the rank four using the method of expanding a determinant to cofactors.
 Solution: Added is third to the second colon. Then, the second row multiplied by -3 is added to the first row. The obtained determinant is then expanded to its cofactors along the second colon: The first colon multiplied by -1 is added to the third colon. The obtained determinant is then expanded along the third colon.   Solved problems contents