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)