Transcript Graphing Systems of Equations

6-1 System of Equations (Graphing):
Step 1: both equations MUST be in slope
intercept form before you can graph the lines
Equation #1: y = m(x) + b
Equation #2: y = m(x) + b
Step 2: find where the line crosses the y-axis (b)
Step 3: determine the slope (m)
m = rise / run
m = y-axis / x-axis
Step 4: graph each equation
6-1 Graphing Possible Solutions:
Only One
Infinite
No Solution
(x,y)
(2 , -1)
Slope-Intercept Form
y = m(x) + b
m = rise / run
m = y-axis / x-axis
y = -3x + 5
y=x-3
+Y
-X
+X
-Y
Graphing
Slope-Intercept Form
y = m(x) + b
m = rise / run
m = y-axis / x-axis
+Y
-X
+X
-Y
Graphing
6-2 Solving Systems (Substitution)
Step 1: Solve an equation to one variable.
Step 2: Use the common variable and substitute
the expression into the other equation.
Step 3: Solve for the only variable left in the
equation to find its value.
Step 4: Plug the new value back into one of the
original equations to find the other value.
3x + y = 6
4x + 2y = 8
3x + y = 6
– 3x
– 3x
y = – 3x + 6
4x + 2y = 8
4x + 2(– 3x + 6) = 8
4x – 6x + 12 = 8
– 12 – 12
– 2x = – 4
– 2x / – 2 = – 4 / – 2
x=2
(2, 0)
3x + y = 6
3(2) + y = 6
–6
–6
y=0
Substitution
POSSIBLE SOLUTIONS
1) Only One (x, y) = crossed lines
2) No Solution (answers don’t equal) = parallel lines
3) Infinite Solutions (answers are equal) = stacked lines
6-3 Elimination (Addition & Subtraction)
• Step 1: Line up the equations so the matching
terms are in line.
• Step 2: Decide whether to add or subtract the
equations to get rid of one variable, then solve.
• Step 3: Substitute the solved variable back into
one of the original equations, then write the
ordered pair (x, y).
Same Signs - SUBTRACT
Opposite Signs + ADD
4x + 6y = 32
Same Signs - SUBTRACT
Opposite Signs + ADD
3x – 6y = 3
(5, 2)
7x + 0 = 35
7x = 35
7x / 7 = 35 / 7
x=5
4 (5) + 6y = 32
20 + 6y = 32
- 20
- 20
6y = 12
6y / 6 = 12 / 6
y=2
Add / Subtract
POSSIBLE SOLUTIONS
1) Only One (x, y) = crossed lines
2) No Solution (answers don’t equal) = parallel lines
3) Infinite Solutions (answers are equal) = stacked lines
6-4 Elimination (Multiplication)
• Step 1: Line up the equations so the matching
terms are in line.
• Step 1.5 (new): Multiply at least one equation
to get two equations containing opposite
terms (example + 6y and – 6y).
• Step 2: Decide whether to add or subtract the
equations to get rid of one variable, then solve.
• Step 3: Substitute the solved variable back into
one of the original equations, then write the
ordered pair (x, y).
5x + 6y = – 8
2x + 3y = – 5
5x + 6y = – 8
2x + 3y = – 5
– 2 (2x + 3y = – 5)
– 4x – 6y = 10
5x + 6y = – 8
– 4x – 6y = 10
x=2
(2, – 3)
2x + 3y = – 5
2 (2) + 3y = – 5
4 + 3y = – 5
–4
–4
3y = – 9
3y / 3 = – 9 / 3
y=–3
Multiplication
POSSIBLE SOLUTIONS
1) Only One (x, y) = crossed lines
2) No Solution (answers don’t equal) = parallel lines
3) Infinite Solutions (answers are equal) = stacked lines