Download - Solving Systems of Equations by Fen Xu and Timothy Lou Ly f(x)= 2x 2 - 2f(x)= -1/2x 2 + 18
SolvingSystems of Equationsby Fen Xu and Timothy Lou Ly
f(x)= 2x2
- 2f(x)= -1/2x2 + 18
The ConceptGraphing a Linear Equation y = 1x
X-Values
Y-Values
-24 -24
-20 -20
-16 -16
-12 -12
-8 -8y = 2x - 4X-
ValuesY-
Values8 12
9 14
10 16
11 18
12 20
(-20, -20)
(-16, -16)
(-8, -8)
(8, 12)
(10, 16)
(12, 20)
The Concept
Dependent System Inconsistent SystemTwo overlapping lines withthe same slope and points
Two lines with the same slopesthat never intersect or share points.
The ConceptThree Ways to Solve
- Graphing
- Addition
- Substitution
Addition
= 3x - 3 + 2y + 4 = 24- 3 + 4
Step 1: Make equations into simplest form of Ax + By = C.
= 3x + 2y = 23
Step 2: Choose to solve for “x” or “y.” For the example,
we’ll solve for “x.”
Multiply by LCD to get rid of fractions
Combine like terms by adding -3 and +4 together
6 ( )_2
3
x - 1 y + 2_+ = 4
x - 2y = 5
simplest formAx + By = C
solve for “x” or “y.”
= 3x + 2y - 1 = 2424Add +1 to both sides to cancel -1 and isolate
variables
AdditionStep 3: Add the two equations so that the y-values
cancel.3x + 2y = 23
+ ( x - 2y = 5)_ Because 2y is being subtracted from 2y, they cancel
4x = 28
Step 4: Continue to solve for “x.”
4x = 28 Divide both sides by 4
x = 7
Add equations so y-values
solve for “x.”
//
4
If you wanted to solve for “y” and cancel“x,” you would need to multiply the 2nd equation by -3
cancel.
AdditionStep 5: We can now put 7 in for “x” in any equation
and find the value of “y.”put 7 in for “x”
find “y.”
x - 2y = 5
= 7 - 2y = 5
= -2y = -2
= y = 1
CHECK:
(7) - 2(1) = 5
(7) - 2 = 5
5 = 5√ √
That’s it! (7, 1) is your solution/intersection!
-2
3x + 2y = 23x- 2y = 5
3(7) + 2(1) = 23
21 + 2 = 23
23 = 23
SubstitutionStep 1: Solve for one of the variables from one equation.Solve for one variables one equation
= 3x + 2y = 233x
= 2y = -3x + 232
Subtract 3x from both sides
Divide the equation by 2
_-32
= y = x 11.5+
Multiply by LCD to get rid of fractions6 ( )
x - 2y = 5_2 3
x - 1 y + 2_+ = 4
= 3x + 2y - 1 = 2424Add +1 to both sides to cancel -1 and isolate variables
= 3x - 3 + 2y + 4 = 24- 3 + 4 Combine like terms by adding -3 and +4 together
_-32x 11.5+
SubstitutionStep 2: Substitute “ ” for “y” in the other
equation: x - 2y = 5 and solve.
x - 2( ) = 5- 2( )
= x + 3x - 23 = 5- 23 5
= 4x = 284
= x = 7
Multiply out -2
Add 23 to both sides
Divide the equation by 4
Step 3: Putting in 7 for “x,” we know that y = 1.
x - 2y = 5= 7 - 2y = 5= -2y = -2= y = 1-2
7 for “x,” y = 1
_-32
y = x 11.5+
X-Values
Y-Values
0 11.5
2 8.5
4 5.5
6 2.5
7 1
X-Values
Y-Values
0 -2.5
2 -1.5
4 -0.5
6 0.5
7 1
_2-3y = x 11.5+
_12
y = x 2.5-
(0, 11.5)
(4, 5.5)
(4, -0.5)
(0, -2.5)
Review
7x-6y=-6
-7x+6y=-4
Try to solve this with the method of your choice:
We’ll check, and if you get it right, you get some candy!(You’ll all get some anyways)