Nash

= = =NASH = =~Solving Systems of Equations~ =

=Vocabulary = =__Systems of Linear Equations__- Two or more linear equations using the same variables and on the same coordinate plane. = =__Solution to a System of Equations__- Any ordered pair in a system that makes the equation true. = =__Substitution Method__- A method of solving a system of equations by replacing one variable with an equivalent expression containing the other variable. = =__Elimination Method__- A method for solving a system of equations. You add or subtract the equations to eliminate a variable. =

Methods for solving systems

 * ** Graphing **
 * ** Substitution **
 * ** Elimination **

=Graphing = =Example: = = Solve by graphing: y=2x-3, y=x-1; Graph both equations on the same coordinate plane. = =y=2x-3 //The slope is 2. The y-intercept is -3.// = =<span style="background-color: #ffffff; color: #404040; font-family: 'Comic Sans MS',cursive; font-size: 70%;">y=x-1 //The slope is 1. The y-intercept is -1.// = =<span style="background-color: #ffffff; color: #404040; font-family: 'Comic Sans MS',cursive; font-size: 70%;">Find the intersection. = =<span style="background-color: #ffffff; color: #404040; font-family: 'Comic Sans MS',cursive; font-size: 70%;">The lines intersect at (2,1), so (2,1) is the solution of the system. = 1=2(2) -3 1=(2)-1 substitute (2,1) for (x,y) 1=4-3 1=2-1 1=1 1=1 ** <span style="color: #00ffff; font-family: 'Comic Sans MS',cursive; font-size: 140%;"> Example: **
 * <span style="background-color: #ffffff; color: #404040; font-family: 'Comic Sans MS',cursive;">Check if the solution, (2,1) makes both equations true. **
 * <span style="background-color: #ffffff; color: #404040; font-family: 'Comic Sans MS',cursive;"><span style="color: #404040; font-family: 'Comic Sans MS',cursive;">y=2x-3 y=x-1
 * Substitution

**<span style="color: #404040; font-family: 'Comic Sans MS',cursive; font-size: 90%;">5x + 7 = 8 5x + 7 - 7 = 8 - 7 ** <span style="color: #404040; display: block; font-family: 'Comic Sans MS',cursive; text-align: left;">**System: y = -4x + 8; y = x + 7 y = -4x + 8 //Start with one equation,// y= x + 7 //then substitute x + 7 for y.// y = y x + 7 = -4x + 8 4x + x + 7 = -4x + 4x + 8 //Solve for x.//

5x = 1 5x = 1 --- --- 5 = 5** <span style="color: #808080; display: block; font-family: 'Comic Sans MS',cursive; text-align: left;">** x = 0.2 //Substitute 0.2 for x to find y using either equation.//

y = x + 7 //Simplify.// y = 0.2 + 7 y = 7.2 **

<span style="color: #404040; display: block; font-family: 'Comic Sans MS',cursive; text-align: left;">**Since x = 0.2 and y = 7.2, the solution is (0.2, 7.2)

Now check the solution using the other equation.

(0.2, 7.2)

y = -4x + 8 7.2 = -4(0.2) + 8 7.2 = -0.8 + 8 7.2 = 7.2 The solution checks.**

<span style="background-color: #ffffff; color: #00ffff; font-family: 'Comic Sans MS',cursive; font-size: 130%;">** Elimination **

<span style="color: #404040; font-family: 'Comic Sans MS',cursive; font-size: 110%;">**Example:** **<span style="font-family: 'Comic Sans MS',cursive; font-size: 110%;">System: 5x - 6y = -32; 3x + 6y = 48 [ Eliminate //y// since the sum of their coefficients is zero.]

Step 1: Add the two equations

5x - 6y = -32 3x + 6y = 48 ** <span style="font-family: 'Comic Sans MS',cursive;">**--- - - -- 8x = 16 //Solve for x// 8x = 16 - -** <span style="color: #404040; font-family: 'Comic Sans MS',cursive;">**<span style="font-family: 'Comic Sans MS',cursive;">8 8 x = 2

Step 2: Solve for //y// using either of the original equations

x = 2 3x + 6y = 48 3(2) + 6y = 48 6 + 6y = 48 //Simplify.//** <span style="color: #404040; font-family: 'Comic Sans MS',cursive;">**6 - 6 + 6y = 48 - 6 6y = 42 6y = 42 - 6 6 y = 7

Since x = 2 and y = 7, the solution is (2, 7)

Step 3: Check: (Use other equation)** <span style="color: #808080; font-family: 'Comic Sans MS',cursive;"> <span style="color: #404040; font-family: 'Comic Sans MS',cursive;">**5x - 6y = -32 //Plug in the variables; x = 2, y = 7.// 5(2) - 6(7) = -32 10 - 42 = -32 -32 = -32 (2,7) is a solution**

<span style="color: #1fe5e5; font-family: 'Comic Sans MS',cursive; font-size: 106%;">Problem Solving
=<span style="color: #404040; font-family: 'Comic Sans MS',cursive; font-size: 76%;">Example: = = = **<span style="color: #404040; font-family: 'Comic Sans MS',cursive;">//A metalworker has some ingots of metal alloy that are 20% copper by weight and others that are 60% copper by weight. How many kg of each type of ingots should the metalworker combine to create an 80 kg of a 52% copper alloy?//

Define: Let x= the mass of the 20% alloy. Let y= the mass of the 60% alloy. Equations: x+y=80, 0.2x+0.6y=0.52(80) Step 1 Choose one of the equations and solve for a variable x+y=80 ** <span style="color: #404040; font-family: 'Comic Sans MS',cursive;">**x+y-y=80-y //solve for x; subtract y from each side// Step 2 Solve for y //(use other equation)// 0.2x+0.6y=0.52(80) 0.2(80-y)+0.6y=0.52(80) //substitute 80-y for x// 16-0.2y+0.6y=0.52(80) //use distributive property// 16+0.4y=41.6 //simplify then solve for y//** 0.4y=25.6 //divide by 0.4// 0.4 0.4** Step 3 Substitute y in another equation to find x. x=80-y x=80-64 //x=16// Step 4 Answer your question in a complete sentence.
 * 16-16+0.4y=41.6-1
 * <span style="color: #404040; font-family: 'Comic Sans MS',cursive;">//y=64//

You need 16kg of 20% copper alloy and 64kg of 60% copper alloy. **

<span style="color: #1fe5e5; display: block; font-family: 'Comic Sans MS',cursive; font-size: 200%; text-align: center;">The END(: