Systems of Equations

Linear equations show progress and starting points.  The starting point being the y-intercept.  Progress being the slope Positive slope would illustrate increase growth Negative slope would show a decrease

Systems of equations use two lines Given two lines there are 3 possibilities  The lines intersect at one point  The lines are the same line  The lines are parallel

Mathematically these calculate to …  The lines intersect at one point a value for x & y (x,y)  The lines are the same line A true mathematical statement 0 = 0  The lines are parallel A false mathematical statement 5 = 0

Calculating Systems  Substitution  Elimination  Graphical Representation (does not really count for accuracy) Either method could be used for all systems of equations. The setup of the original problem lends itself to one calculation or the other.

Substitution – use substitution when…  One of the equations is already solved for a variable. y = 2x – 5 3x + 4y = 13 Substitute the first equation into the second 3x + 4(2x – 5) = 13 Solve for the variable 3x + 8x – 20 = 13 11x = 33 x = 3 Substitute back into one of the original equations y = 2(3) – 5 = 1 Final Answer (3,1)

Substitution – use substitution when…  One of the variables is EASY to solve for. 3x + 4y = 13 x – 2y = 6 Solve for x in the second equation x = 2y + 6 Continue like the previous problem - Substitute 3x + 4(2y + 6) = 13 3x + 8y + 24 = 13 11x = -11  x = -1 Substitute back into one of the original equations 3(-1) + 4y = 13  y = 13 4y = 16  y = 4 Final Answer (-1,4)

Elimination – use elimination when substitution is not set up.  Elimination ELIMINATES a variable through manipulating the equations.  Some equations are setup to eliminate.  Some systems only one equation must be manipulated  Some systems both equations must be manipulated

Setup to Eliminate  Given 2x – 4y = 8 3x + 4y = 2  The y terms are opposites, they will eliminate  Add the two equations 5x = 10  x = 2  Substitute into an original equation 3(2) + 4y = 2  6 + 4y = 2  4y = -4  y = -1 Final Answer (2,-1)

Manipulate ONE eqn. to Eliminate  Given 2x + 2y = 8 3x + 4y = 2  Multiply the first equation by – 2 to elim. y terms -4x – 4y = -16 3x + 4y = 2  Add the two equations -1x = -14  x = 14  Substitute into an original equation 3(14) + 4y = 2  y = 2  4y = -40  y = -10 Final Answer (14,-10)

Manipulate BOTH eqns. to Eliminate  Given 2x + 3y = 4 3x + 4y = 2  Multiply the first equation by 3 & the second equation by -2 to elim. x terms 6x + 9y = 12 -6x - 8y = -4  Add the two equations y = 8  Substitute into an original equation 2x + 3(8) = 4  2x + 24 = 4  2x = -20  x = -10 Final Answer (-10,8)

Assignment Online – 41 Systems