Chapter 2 Solving Linear Equations

Mathematically Speaking 15x + 13y – 4(3x+2y) 15x + 13y – 12x - 8y 15x – 12x + 13y - 8y (15 – 12)x + (13 – 8)y 3x + 5y Can you identify what happens in each step?

Can you identify what has happened in each step? 15x + 13y – 4(3x+2y) 15x + 13y – 12x - 8y 15x – 12x + 13y - 8y (15 – 12)x + (13 – 8)y 3x + 5y - Given -Distributive -Commutative -Factor -Addition

Identify the steps used to solve the equation, m + 4 = 29. m+4=29 - 4=-4 m =25 Given Inverse +  - Evaluate

Identify the steps used to solve the equation. 3x + 4 = = - 4 3x = 15  3=  3 x = 5 Given Inverse +   Evaluate Inverse *   Evaluate

Identify the steps used to solve the equation. 5x – 4 = 2(x – 4) x – 4 = 2x – x – 4 = 2x x = 14

Identify the steps used to solve the equation. 5x – 4 = 2(x – 4) x – 4 = 2x – x – 4 = 2x x = 14 Given Distributive Addition Inverse Ops Like terms Inverse Ops

Identify the steps used to solve the equation. -5x x = 7x – 8 + 9x -3x +3 = 16x = 19x

Identify the steps used to solve the equation. -5x x = 7x – 8 + 9x -3x +3 = 16x = 19x Like Terms Inverse Ops Symmetric property Given

So what is the definition? Which of these equations are linear? x+y = 5 2x+ 3y = 4 7x-3y = 14 y = 2x-2 y=4 x 2 + y = 5 x = 5 xy = 5 x 2 +y 2 = 9 y = x 2 3 y Linear Not Linear The degree must be one.

2.1 What is a solution? What happens when one solves an equation? You might say “One gets an answer.” What is the format of that answer?

What happens when one solves an equation? 1.The solutions is a Unique solution. 2.The solution is Infinite solutions. 3.The is no possible solution.

What happens when one solves an equation? 1.The solution is a Unique solution. There is only ONE numerical answer to solve the equation. 2.The solution is Infinite solutions. IDENTITY. The equations are mathematically equivalent. 3.There is no possible solution. INCONSISTENT. With linear equations this means there is no point of intersection.

2.2 One linear equation in one variable

One Solution. 3x + 4 = = - 4 3x = 15  3=  3 x = 5

Infinite Solutions. IDENTITY x – 4 = (x + 4x) x – 4 = 5x – x + 10 = 5x = 10

No Solution. INCONSISTENT -7x x = 2x – 8 - 8x -6x +3 = -6x -8 3 = -8

2.3 Several linear equations in one variable

Systems of Equations Solving systems of equations with two or more linear equations Substitution Elimination Cramer’s Rule Graphical Representation

The 3 possible solutions still occur. 1.The solution is a Unique solution. This one solution is in the form of a point. (e.g. (x,y), (x,y,z) ) 2.The solution is Infinite solutions. IDENTITY. The lines are the same line. 3.There is no possible solution. INCONSISTENT. The lines are parallel (2-D) or skew (3-D).

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)

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)

Identity Example 2x + 3y = 12 y = -2/3 x + 4 Using substitution 2x + 3(-2/3 x + 4) = 12 2x – 2x + 12 = = 12 Identity

Inconsistent Example 3x – 4y = 18 3x – 4y = 9 Use Elimination by multiplying Eqn 2 by -1. 3x – 4y = 18 -3x + 4y = -9 0 = 9 False Inconsistent

3 Equations: 3 Variables required Eqn1: 3y – 2z = 6 Eqn2: 2x + z = 5 Eqn3: x + 2y = 8 Solve Eqn2 for z z = -2x + 5 Now substitute into Eqn1 3y – 2(-2x+5) = 6 3y + 4x – 10 = 6

3 Equation continued… NEW: 4x + 3y = 16 Eqn3: x + 2y = 8 Now one can either substitute or eliminate NEW: 4x + 3y = 16 Eqn3( *-4 ): -4x - 8y = y = -16 y = 16/5

Now having a value for y, one can substitute into x + 2(-16/5) = 8 x = /5 = 40/5 + 32/5 x = 72/5 This can now be substituted into our Eqn2 solved for z z = - 2(72/5) + 5 z = -144/5 + 5 = -144/5 + 25/5 z = -119/5 Final Answer (72/5, -16/5, -119/5) And still continued…

Matrices: Cramer’s Rule Dimensions: row x columns Determinant abcdabcd ad - bc efef

Cramer’s Rule set up ebfdebfd aebfaebf x =y = determinant

Example 2x + 3y = 5 4x + 5y = 7 The determinant is = x = y = -2

Solve for x and y… 4/-2 x = -2 -6/-2 y = x setupy setup -2 Final answer (-2,3)

You cannot use Cramer’s Rule if the difference of the products is 0.

Verbal Models Verbal Models are math problems written in word form General Rule: Like reading English - Left to Right Special Cases: Change in order terms some time called “turnaround” words (Cliff Notes: Math Word Problems, 2004)

Convert into Math… Two plus some number A number decreased by three Nine into thirty-six Seven cubed Eight times a number Ten more than five is what number 2+x x-3 36 / 9 7^ x = x into more than

MORE Convert into Math… Twenty-five percent of what number is twenty- two? The quantity of three times a number divided by seven equals nine. The sum of two consecutive integer is * x = 22 (3x)/7 = 9 x + (x+1) = 23

Work Problem. I can mow the yard in 5 hours. My husband can mow the yard in 2 hours. If we mowed together how long would it take for us to mow the yard.

Solution My rate is 1 yard per 5 hours: 1/5 t Doug’s rate is 1 yard per 2 hours; ½ t together = addition The whole job = 1 the common denominator is 10 Solve for t 7t = 10; t = 10/7 or ish

Formulas you should know… Area of Rectangle Perimeter of Rectangle Area of Triangle Area of Circle A = hb P = 2 (h + b) A = ½ hb A =  r 2

Candy I bought 3 bags of candy and 5 chocolate bars. I spent $13. My friend spent $17 and she bought 4 bags of candy and 6 chocolate bars. What is the cost of the candy bags and chocolate bars?

Solution 3b + 5c = 13 4b + 6c = 17 det = = -2 x = y = x = (78-85)/-2y = (51-52)/-2 x = -7/2 = $3.50y = -1/-2 =