6-4 Elimination Using Multiplication
SWABT: Solve a system of equations using elimination by multiplication.
Important concepts:
Understand exact opposites and eliminating a variable
Understand solving an equation
Understand a system of equations
Step 1: Determine which variable to eliminate first
Step 2: Multiply one or both equations to obtain exact opposites for the
variable you wish to eliminate first.
Step 3: Add the equations, eliminating one variable.
Step 4: Solve the resulting equation for one variable.
Step 5: Write one of the original equations.
Step 6: Substitute the solution you found in step 4 into one of the original equations.
Step 7: Solve for the other variable.
Step 8: Write the solution as an ordered pair.
Step 9: Check
Example 1:
5x + 6y = -8
2x + 3y = -5
Step 1: I notice that 3y is a factor of 6y…making the “y”
in this case
5x + 6y = -8
-2(2x + 3y = -5)
easier to eliminate
5x + 6y = -8 Step 2:Multiply 2nd equation by -2
+( -4x -6y = 10)
x
=2
Step 3 & 4: Add & Solve for “x”
5x + 6y = -8
5(2) + 6y = -8
10 + 6y = -8
-10
-10
6y = -18
/6 /6
y = -3
(x, y) (2, -3)
Step 5: Rewrite one of original equations
Step 6: Substitute value found for x into original equation
Step 7: Solve for “y”
Check:
Step 9
Step 8: write the solution as an ordered pair
Example 2:
6x – 2y = 10
3x – 7y = -19
Step 1: I notice that 3x is a factor of 6x…making the “x”
6x - 2y = 10
-2(3x - 7y = -19)
6x - 2y = 10
6x – 2(4) = 10
6x - 8 = 10
+8 +8
6x = 18
/6 /6
x=3
(x, y) (3, 4)
easier to eliminate in this case
6x - 2y = 10 Step 2:Multiply 2nd equation by -2
+( -6x +14y = 38)
12y = 48
Step 3: Add
/12 /12
Step 4: Solve for “y”
y=4
Step 5: Rewrite one of original equations
Step 6: Substitute value found for “y” into original equation
Step 7: Solve for “x”
Step 8: write the solution as an ordered pair
Now You Try:
Example 3:
9r + q = 13
3r + 2q = -4
Step 1: I notice that q is a factor of 2q…making the “q”
___(9r + q = 13)
3r + 2q = -4
9r + q = 13
9(___) + q = 13
18 + q = 13
_____________
q = ___
(q, r) (___,___)
easier to eliminate in this case
-18r -2q = -26 Step 2:Multiply 1st equation by -2
+( ______________)
-15r = -30
Step 3: _________
/-15 /-15
Step 4: Solve for “___”
r = _____
Step 5: Rewrite one of original equations
Step 6: Substitute value found for “___” into original equation
Step 7: Solve for “____”
Step 8: write the solution as an ordered pair
Sometimes you must Multiply BOTH equations:
Example 4:
4x + 2y = 8
3x + 3y = 9
-3(4x + 2y = 8)
4(3x + 3y = 9)
4x + 2y = 8
4x +2(2) = 8
4x + 4 = 8
-4 -4
4x = 4
/4 /4
x=1
(x,y) (1, 2)
-12x - 6y = -24
12x + 12y = 36
6y = 12
Step 1: eliminate “x”
Step 2: Mult 1st equation by -3, 2nd
equation by 4
Step 3: Add equations
Step 4: Solve for “y”
/6 /6
y=2
Step 5: Rewrite one of original equations
Step 6: Substitute y =2
Step 7: Solve for “x”
Step 8: write the solution as an ordered pair
Example 5:
6a + 2b = 2
4a + 3b = 8
-2(6a + 2b = 2)
3(4a + 3b = 8)
6a + 2b = 2
6a + 2(4) = 2
6a + 8 = 2
-8 -8
6a = -6
/6 /6
a = -1
(a,b)
-12a -4b = -4
12a + 9b = 24
5b = 20
/5
/5
b=4
Step 1: eliminate “a”
Step 2: Mult 1st equation by -2, 2nd
equation by 3
Step 3: add equations
Step 4 Solve for “b”
Step 5: Rewrite one of original equations
Step 6: Substitute b = 4
Step 7: Solve for “a”
Step 8: Write the solution as an ordered pair
Now You Try:
Example 6:
2a + 5b = -10
5a – 3b= 6
__(2a + 5b = -10)
__(5a -3b = 6)
2a + 5b = -10
2(___) + 5b = -10
________________
_______________
/5 /5
b = ______
(a,b) (________,_______)
________________
Step 1: eliminate “____”
________________
Step 2: Mult 1st equation by___, 2nd
___________
equation by____
/31
/31 Step 3: add equations
a = ____
Step 4 Solve for “___”
Step 5: Rewrite one of original equations
Step 6: Substitute a = 0
Step 7: Solve for “b”