Review:

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1. Review How to graph an equation
Step 1 Setup x/y table and a set of coordinate axes
Step 2 The first point is the Y-intercept with coordinates (0, b). Plot on graph.
Step 3 For second point, EITHER Pick any x-value and plug-in to solve for y-value. Find X-intercept where y0
Step 4 Plot second point and connect with Y-Intercept to form a line
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2. What is a System of Equations?

• A System of Equations is a pair of equations with
  two variables
• A system of equations has only one solution (or
  sometimes no solution)
• The solution is where the lines intersect

Example 1 Example 2 y 2x 4 x
y 3 y 5x 13 3x 5y 17
Slope-Int Form
Standard Form
Slope-Int Form
Standard Form
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• You can find the solution (solve) to a system of
  equations by graphing

3. Finding Solution to Systems of Equations By
Graphing
Step 1 Graph both equations
Step 2 The solution is the point (x, y) where the lines intersect
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3. Finding Solution to Systems of Equations By
Graphing Example 1
y x 3 (Equation 1) y -x 5 (Equation 2)
Equation 1
Equation 2
X Y
0

X Y
0

-3
5
2
-1
2
3
y 0 3 -3
y 0 5 5
y 2 3 -1
y -2 5 3
5
Equation 1
Equation 2
3. Finding Solution to Systems of Equations By
Graphing Example 1
X Y
0

X Y
0

-3
5
2
-1
2
3

• Make sure to EXTEND YOUR LINES

• The solution is ( 4 , 1 )

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3. Finding Solution to Systems of Equations By
Graphing Example 2
y 3x 5 (Equation 1) y -2x 10 (Equation 2)
Equation 1
Equation 2
X Y
0

X Y
0

5
10
-1
2
2
6
y 3(0)5 5
y -2(0)10 10
y 3(-1)5 2
y -2(2)10 6
7
Equation 1
Equation 2
3. Finding Solution to Systems of Equations By
Graphing Example 2
X Y
0

X Y
0

5
10
-1
2
2
6

• Make sure to EXTEND YOUR LINES

• The solution is ( 1 , 8 )

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4. Finding Solution Systems of Equations By
Substitution Both Equations Slope-Intercept

• If two things equal the same value or variable,
  we can set them equal to each other

• Example 1
• 5 2 3
• 5 4 1
• 5 5
• 2 3 4 1

• Example 2
• y 2x 3
• y x 1
• y y
• 2x 3 x 1

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• Finding Solution to System of Equations
• Set equations equal to each other and solve for x
• This is called substitution since we are
  substituting in one equation for y

4. Finding Solution Systems of Equations By
Substitution Both Equations Slope-Intercept
I DO
y 2x 3 y x 1
2x 3 x 1
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I DO
4. Finding Solution Systems of Equations By
Substitution Both Equations Slope-Intercept

• Now, solve for x

2x 3 x 1
- x -x .
x 3 1
- 3 -3
x -2
Check 2(-2) 3 (-2) 1 -4 3 -2 1
-1 -1
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• Remember, a solution is (x, y), so you need BOTH
  COORDINATES
• Check your solution by plugging both (x, y) into
  the OTHER equation

I DO
4. Finding Solution Systems of Equations By
Substitution Both Equations Slope-Intercept
x -2
Check y x 1
y 2(x) 3 y 2(-2) 3 -1 Solution
(-2, -1)
-1 -2 1 -1 -1
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Set them equal and solve for x
WE DO
3x 2 -2x 7
4. Finding Solution Systems of Equations By
Substitution Both Equations Slope-Intercept
2x 2x .
Solve y 3x 2 y -2x 7
5x 2 7
- 2 -2
5x 5
5 5
Plug-in x and solve for y y 3(x) 2 y
3(1) 2 5 Solution (1, 5)
x 1
Check 3(1) 2 -2(1) 7 3 2 -2 7
5 5
Check 5 -2(1) 7
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Equation 1 y 3x2
Equation 2 y -2x 7
4. Finding Solution Systems of Equations By
Substitution Both Equations Slope-Intercept
X Y
0 2
1 5
X Y
0 7
1 5

• The solution is ( 1 , 5 )

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YOU DO
4. Finding Solution Systems of Equations By
Substitution Both Equations Slope-Intercept
1 y 3x 4 y 2x 2
2 y 5x 8 y -3x 16
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5. Finding Solution Systems of Equations By
Substitution 1 Equation Slope Int. 1 Equation
Standard Form

• We can use substitution to solve this type of
  system
• Substitute or plug in for y in the standard form
  equation

y 2x 3 3x 2y 13
Slope-Int Form
Standard Form
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I DO
y 2x 3 3x 2y 13
5. Finding Solution Systems of Equations By
Substitution 1 Equation Slope Int. 1 Equation
Standard Form
Slope-Int Form
Standard Form
1. Substitute or Plug in the first equation
into the second equation
2. Solve for X
3x 2(2x 3) 13
3x 4x 6 13
7x 6 13
-6 -6
7x 7
7 7
y 2(x) 3 y 2(1) 3 5 Solution (1, 5)
x 1
3. Plug in and solve for y
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WE DO
y 2x 3 7x 1y 7
5. Finding Solution Systems of Equations By
Substitution 1 Equation Slope Int. 1 Equation
Standard Form
Slope-Int Form
Standard Form
1. Substitute or Plug in the first equation
into the second equation
2. Solve for X
7x 1(2x 3) 7
7x 2x 3 7
5x 3 7
3 3
5x 10
5 5
y 2(x) 3 y 2(2) 3 7 Solution (2, 7)
x 2
3. Plug in and solve for y
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YOU DO
5. Finding Solution Systems of Equations By
Substitution 1 Equation Slope Int. 1 Equation
Standard Form
1 y x 6 x y -2
2 9x 2y 3 y 3x 6