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Section 2.1 Solving Systems of Equations
VocabularySystem of Equations - A set of two or more
equations Solution – An ordered pair representing the
solution common to both equations in a system
Consistent System – Has at least 1 solutionIndependent system – Has exactly 1 solutionInconsistent system – There is no solution
Solve by GraphingEquations must be in y- intercept form ( y = mx
+ b) where m is the slope and b is the y – intercept
The Y variable cannot be negativeFor Example3x – 2y = -6 becomes y = 3/2x + 3x + y = -2 becomes y = -x -2When the graphs. . .Intersect, there is one solutionAre on the same line, there are many solutions Are parallel, there are no solutions *graphs
with the same slope and different intercepts will always be parallel*
Solve by Elimination
In elimination one variable is eliminated so that the equation can be solved in terms of the other
5(1.5x + 2y = 20) BECOMES 7.5x + 10y = 100 The 10y & - 10y will cancel
2(2.5x – 5y = -25) 5x – 10y = - 50 out leaving only the x variable
This leaves 12.5x = 50 which means x = 4Once an answer is reached for one variable you
can plug it into any of the original equations to find the second variable
1.5x + 2y = 20 ====> 1.5(4) + 2y = 20 ====> 2y = 14 =====> y = 7
Solve by Substitution
In substitution one of the equations is set equal to a variable and substituted into the second equation
For Example2x + 3y = 8x- y = 2 becomes x = y + 2 After
Substitution 2(y + 2) + 3y = 8 ==> 5y = 4 ==> y = 4/5
Once a solution is reached it can be plugged into any of the other original equations
x – y = 2 ===> x – 4/5 = 2 ===> x = 14/5
HomeworkPG 71 #’s 21-2421. 5x –y = 16 22. 3x -5y = -8
23. y = 6 – x 2x+3y = 3 x + 2y = 1 x = 4.5 + y
24. 2x +3y = 312x -15y = -4
Solve By Elimination
Elimination with three variables works the same as with two, your goal is to eliminate one variable at a time.
For Examplex – 2y + z = 15 Choose one variable
and equation2x + 3y – 3z = 1 to use for the
remaining two equations. For this example4x + 10y – 5z = -3 I have chosen the
equation in bold.
-2 (x – 2y + z = 15) and -4(x – 2y + z = 15)
2x +3y -3z = 1 4x + 10y – 5z = -3
-2x + 4y – 2z = -30 (2x’s cancel out) -4x +
8y – 4z = -60 (4x’s cancel out) 2x + 3y -3z = 1 4x +10y –
5z = -3
7y – 5z = -29 *Now use the elimination method with these two equations* 18y + 9z = -63
-9(7y – 5z = -29) ===> -63y +45z = 261 (45z’s cancel out) 5(18y – 9z = -63) ===> 90y – 45z = -315 Leaving Only
===> 27y = -54 Therefore y = -2 Substitute Y into one of the previous equations 7(-2) – 5z = -29
Z = 3 Then go back to one of the original equations and plug in the
value of Z and Y to find X X – 2(-2) +3 = 15 X = 8 Solution x = 8, y = -2, z = 3
Solve by Substitution The same problem can be solved by using the substitution method x – 2y + z = 15 becomes x = 2y – z + 15 now substitute it for both
remaining equations 2x + 3y – 3z = 1 4x + 10y – 5z = -3 2(2y – z + 15) +3y – 3z = 1 and 4(2y
– z +15)+10y – 5z = -3 4y – 2z +30 +3y -3z = 1 8y – 4z +60
+10y -5z = -3 7y – 5z = -29 Now use substitution with these two equations 18y –
9z = -63 18y – 9z = -63 ===> z = 2y + 7 7y – 5z = -29 7y -5(2y+7) = -29 7y – 10y – 35 = -29 y = -2 Now substitute into the previous equation 7(-2)
– 5z = -29 ==> z = 3 Finally substitute the values for z and y in any original equation X – 2(-2)+3 = 15 ==> x = 8 Solution x = 8, y = -2, z = 3
Homework Pg 74 4-6
Solve each system of equations using the substitution method or elimination method
4. 4x +2y +z = 7 5. x – y – z = 7 6. 2x – 2y +3z = 6
2x +2y – 4z = -4 –x +2y – 3z = -12 2x – 3y +7z = -1
X +3y – 2z = -8 3x – 2y +7z = 30 4x – 3y +2z = 0
Vocabulary 1 .Matrix- rectangular array of terms.
2. Element- the terms in the matrices.3. M x N Matrix- a matrix with “m” rows and “n” columns.4. Dimensions- are the “m” and “n”, or the rows and columns.5. Row Matrix- a matrix that has only one single row.6. Column Matrix- a matrix with only one single column.7. Square Matrix- has the same number of rows as columns.8. Nth order- is when “n” is the number of rows and columns in a matrix.9. Zero Matrix- is
2.5 Determinants and Multiplicative Inverses of Matrices
VocabularyDeterminant- of Minor- an element of any nth-order
determinant is a determinant of order (n-1).Identity matrix for multiplication- for any
square matrix A is the matrix I, such as that AI=A and IA=A. Represented as
Inverse matrix- can be designated as
2.6 Solving Systems of Linear
InequalitiesFind the maximum and minimum values
of f(x, y) x – y= 2 for the polygonal convex set determined by the system of inequalities.