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Linear equations

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Linear equations

  1. 1.  A pair of linear equations in two variables is said to form a system of simultaneous linear equations.  For example: 2x – 3y +4 = 0 and x + 7y -1 = 0  These two equations form a system of two linear equations in variables x and y.
  2. 2.  The general form of a linear equation x and y is  Ax + by +c = 0, where a and b is not equal to zero and are real numbers.  A solution of such an equation is a pair of values. One is for x and the other for y. Once the values of x and y are represented the two sides of the equation hold for them to be equal.  Every linear equation in two variables has infinitely many solutions which can be represented on a certain line.
  3. 3.  Let us consider the following system of two simultaneous linear equations in two variable:  2x-y=-1 and 3x+2y=9  Here we assign any value to one of the two variables and then determine the value of the other variable from the given solution by using a t-chart
  4. 4. For the equation:  2x-y=-1 solve for y y=2x+1 Plug x=0,2 to get y values  3x+2y=9 solve for y y=9-3x Plug x=3,-1 to get y values Any questions on how to Complete t-chart? x 0 2 y 1 5 x 3 -1 y 0 6
  5. 5. X Y 0 1 2 5 X Y 3 0 -1 6 Y=2x+1 Y=9-3x 0 1 2 3 4 5 6 7 -2 -1 0 1 2 3 4 Y-Values
  6. 6.  The most common used algebraic methods of solving simultaneous linear equations in two variables are:  Method of by substitution.  Method by equating the coefficient.  Method by elimination
  7. 7.  Solve the equations given by solving for x and substitute the x-value of (i) first equation into (ii) second equation to get one equation (i) x+2y=-1 and (ii)2x-3y=12 (i): x=-2y-1 plug this equation into (ii) by substituting value of x 2(-2y-1)-3y=12 distribute and simplify -4y-2-3y=12 combine like terms and solve for y -7y-2=12 -7y=14 y=-2 putting the value of y in equation (i) w get x=-2y-1 x=-2(-2)-1 x=3 Hence solution of the equation is (3,-2)
  8. 8.  Try on your own  Use previous slide to solve for x and y in these two equations given:  y=5x-1 and 2y=3x+12
  9. 9.  y=5x-1 (i) and 2y=3x+12 (ii) Step 1: substitute (i) into (ii) from y value. Step 2: 2(5x-1)=3x+12 distribute Step 3: 10x-2=3x+12 combine like terms to solve for x Step 4: 7x-2=12 7x=14 x=2 Step 5: Use value of x to solve for y in (i) equation Step 6: y=5(2)-1 y=9 Step 7: Solution: x=2 and y=9, check your work by plugging in these values in both equations to make sure left equals right in both equations. Any questions?
  10. 10.  In this method, we eliminate one of the two variable which can easily be solved.  Putting the value of this variable in any of the given equations. The value of the other variable can be obtained.  For example we want to solve:  3x+2y=11 and 2x+3y=4
  11. 11.  (i) 3x+2y=11 and (ii) 2x+3y=4  First we can use a method by equating the coefficient which means multiplying (i) by 3 and (ii) by -2.  We get 3(3x+2y=11)------9x+6y=33 -2(2x+3y=4)------- -4x-6y=-8 Now we can eliminate y values by adding both (i) and (ii) 9x+6y=33 + -4x-6y=-8 ---------------- 5x=25 solve for x x=5
  12. 12.  Putting the value of y in equation (i)  3x+2y=11  3(5)+2y=11  15+2y=11  2y=-4  y=-2  Solution: x=5 and y=-2  Check solutions
  13. 13.  Try on your own  Use previous slides to solve for x and y in these two equations given:  x+3y=-5 and 4x-y=6
  14. 14.  x+3y=-5 (i) and 4x-y=6 (ii)  Step 1: Multiply (i) by -4 to eliminate x.  Step 2: -4(x+3y=-5)------- -4x-12y=20  Step 3: Add both equations and combine like terms to eliminate x.  Step 4: 4x-y=6 + -4x-12y=20 ---------------------- -13y=26 -y=2…..multiply by (-1)….. y=-2 Step 5: Plug in y=-2 into equation (i) x+3(-2)=-5…. x-6=-5 x=1 Step 6: Check solutions
  15. 15.  Given two equations, with two variables, we are then able to use different methods to solve for x and y  Once we found the values of x and y, are we finished?  We need to make sure both values hold for both equations.  What are the two methods we went over?  The methods are methods by substitution and method by elimination.  Any questions?

Hinweis der Redaktion

  • Introduce video to introduce two equations to solve for two variables
  • Given two equations, introduce the t-chart to graph the equation
  • Solve for y in both equations and plug in value for first equation x=0 to get y value and plug in x=2 to get second y value. Have students work on second equation plug in x=3,-1. Give students 2 minutes to solve for variables.
  • Have students graph the equation with the given values before introducing the graph.
  • Introduce the methods to the students
  • After students found the solution of the two values for x and y, have students plug those values in to show both sides of both equations hold and are equal.
  • Given students about 5 minutes to solve for x and y. walk around classroom to see who needs help.
  • Given students about 5 minutes to solve for x and y. walk around classroom to see who needs help.

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