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 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.
 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.
 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
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
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
 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
 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)
 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
 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?
 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
 (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
 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
 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
 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
 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?

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Linear equations

  • 1.
  • 2.
  • 3.  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.
  • 4.  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.
  • 5.  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
  • 6. 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
  • 7. 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
  • 8.  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
  • 9.  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)
  • 10.  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
  • 11.  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?
  • 12.  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
  • 13.  (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
  • 14.  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
  • 15.  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
  • 16.  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
  • 17.  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?

Editor's Notes

  1. Introduce video to introduce two equations to solve for two variables
  2. Given two equations, introduce the t-chart to graph the equation
  3. 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.
  4. Have students graph the equation with the given values before introducing the graph.
  5. Introduce the methods to the students
  6. 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.
  7. Given students about 5 minutes to solve for x and y. walk around classroom to see who needs help.
  8. Given students about 5 minutes to solve for x and y. walk around classroom to see who needs help.