Systems

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Systems

  1. 2. <ul><li>You can open your book to page 417 to refresh your memory as you start your warm-up. The information presented in this lesson will be included in sections 9-2, 9-4, and 9-5. </li></ul>
  2. 3. <ul><li>Please take 5 minutes to work on the following problems… </li></ul><ul><li>1. x+y= 15 </li></ul><ul><li>4x+3y= 38 </li></ul><ul><li>2. 6c+7d= -15 </li></ul><ul><li>6c-2d= 12 </li></ul><ul><li>3. 3a+4b= 2 </li></ul><ul><li>5a+9b= 1 </li></ul>
  3. 4. <ul><li>Solve one equation for one of the variables </li></ul><ul><li>Substitute this expression in the other equation and solve for the other variable </li></ul><ul><li>Substitute this value in the equation in step 1 and solve </li></ul><ul><li>Check the values in both equations </li></ul>
  4. 5. <ul><li>Example 1: 2x-3y=4 </li></ul><ul><li>x+4y= -9 </li></ul><ul><li>1.) 2x-3y=4 </li></ul><ul><li>x= -9-4y </li></ul><ul><li>2.) 2(-9-4y)-3y=4 </li></ul><ul><li>y= -2 </li></ul><ul><li>3.) x+4(-2)= -9 </li></ul><ul><li>x= -1 </li></ul><ul><li>4.) 2(-1)-3(-2)=4 4=4 yes </li></ul><ul><li>(-1)+4(-2)= -9 -9= -9 yes </li></ul><ul><li>Answer (-1,-2) </li></ul>Use the 4 step process… Solve one equation Substitute and solve in other equation Substitute this value in the equation in step 1 and solve Check
  5. 6. <ul><li>Example 2: 2x-8y=6 </li></ul><ul><li>x-4y=8 </li></ul><ul><li>1.) 2x-8y=6 </li></ul><ul><li>x=4y+8 </li></ul><ul><li>2.) 2(4y+8)-8y=6 </li></ul><ul><li>16=6 no </li></ul><ul><li>3.) No solution; special case where the variables cancel each other out and the 4 step process does not work. </li></ul>Use the 4 step process… Solve one equation Substitute and solve in other equation Substitute this value in the equation in step 1 and solve Check
  6. 7. <ul><li>Add or subtract the equation to eliminate one variable </li></ul><ul><li>Solve the resulting equation for the other variable </li></ul><ul><li>Substitute in either original equation to find the value of the first variable </li></ul><ul><li>Check in both equations </li></ul>
  7. 8. <ul><li>Example 1: 2c+3d=0 </li></ul><ul><li> + 5c-3d=21 </li></ul><ul><li>1.) 7c = 21 </li></ul><ul><li> 7 7 </li></ul><ul><li>2.) c=3 </li></ul><ul><li>3.) 2(3)+3d=0 </li></ul><ul><li>6+3d=0 </li></ul><ul><li>d= -2 </li></ul><ul><li>4.) 2(3)+3(-2)=0 0=0 yes </li></ul><ul><li>5(3)-3(-2)=21 21=21 yes </li></ul><ul><li>Answer (3,-2) </li></ul>Add or subtract the equation to eliminate one variable Solve the resulting equation for the other variable Substitute in either original equation to find the value of the first variable Check in both equations
  8. 9. 5 Step Addition and Subtraction Process Using Multiplication Multiply one or both the equations so the absolute value of the coefficients of one of the variables are equal Add or subtract the equation to eliminate one variable Solve the resulting equation for the other variable Substitute in either original equation to find the value of the first variable Check in both equations <ul><li>Example 2: 4x+5t=22 </li></ul><ul><li>5x-t= 13 </li></ul><ul><li>1.) 4x+5t=22 </li></ul><ul><li> (5)5x-t= 13(5) </li></ul><ul><li>2.) 4x+5t=22 </li></ul><ul><li> + 25x-5t=65 </li></ul><ul><li>3.) 29x = 87 </li></ul><ul><li> 29 29 </li></ul><ul><li> x=3 </li></ul><ul><li>4.) 5(3)-t=13 </li></ul><ul><li>t=2 </li></ul><ul><li>5.) 4(3)+5(2)=22 22=22 yes </li></ul><ul><li> 5(3)-2=13 13=13 yes </li></ul><ul><li>Answer (2,3) </li></ul>
  9. 10. <ul><li>Example 3: 6n+8c-4=0 </li></ul><ul><li>9n+10c-7=0 </li></ul><ul><li>1.) (3/2)6n+8c-4=0(3/2) </li></ul><ul><li>9n+10c-7=0 </li></ul><ul><li>2.) 9n+12c-6=0 </li></ul><ul><li>-( 9n+10c-7=0) </li></ul><ul><li>2c = -1 </li></ul><ul><li>3.) 2 2 </li></ul><ul><li>c= -1/2 </li></ul><ul><li>4.) 6n+8(-1/2)-4=0 </li></ul><ul><li>n= 4/3 </li></ul><ul><li>5.) 6(4/3)+8(-1/2)-4=0 0=0 yes </li></ul><ul><li>9(4/3)+10(-1/2)-7=0 0=0 yes </li></ul><ul><li>Answer (-1/2, 4/3) </li></ul>5 Step Addition and Subtraction Process Using Multiplication Multiply one or both the equations so the absolute value of the coefficients of one of the variables are equal Add or subtract the equation to eliminate one variable Solve the resulting equation for the other variable Substitute in either original equation to find the value of the first variable Check in both equations
  10. 11. <ul><li>Please take 5 minutes to complete your assessment… </li></ul><ul><li>y/2= 2-x </li></ul><ul><li>6x+3y= 12 </li></ul><ul><li>5x-y= 12 </li></ul><ul><li>3x+y= 4 </li></ul><ul><li>4x-5y= 23 </li></ul><ul><li>3x+10y= 31 </li></ul>
  11. 12. <ul><li>-Remember… </li></ul><ul><li>-To solve any system you must find the value of one variable </li></ul><ul><li>-After that you substitute that answer in one equation to find the other variable </li></ul><ul><li>-Finally you check your answers </li></ul><ul><li>(Important to have on foldable) </li></ul>

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