Solving Linear Systems by Graphing

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• Solving Linear Systems by Graphing

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Focus 5 Learning Goal (HS.A-CED.A.3,
HS.A-REI.C.5, HS.A-REI.C.6, HS.A-REI.D.11,
HS.A-REI.D.12) Students will write, solve and
graph linear systems of equations and
inequalities.
4 3 2 1 0
In addition to level 3.0 and above and beyond what was taught in class,  the student may Make connection with other concepts in math Make connection with other content areas. The student will write, solve and graph systems of equations and inequalities. - Solve systems of linear equations graphically, with substitution and with elimination method. - Solve systems that have no solutions or many solutions and understand what those solutions mean. - Find where linear and quadratic functions intersect. - Use systems of equations or inequalities to solve real world problems. The student will be able to - Solve a system graphically. - With help the student will be able to solve a system algebraically. With help from the teacher, the student has partial success with solving a system of linear equations and inequalities. Even with help, the student has no success understanding the concept of systems of equations.
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• With an equation, any point on the line (x, y) is
  called a solution to the equation.
• With two or more equations, any point that is
  true for both equations is also a solution to the
  system.

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Is (2,-1) a solution to the system?

• 3x 2y 4

1. Check by graphing each equation. Do they cross at
   (2,-1)?

-x 3y -5
2. Plug the (x,y) values in and see if both
equations are true.
3(2) 2(-1) 4 6 (-2) 4 4 4
-2 3(-1) -5 -2 (-3) -5 -5 -5
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Helpful to rewrite the equations in
slope-intercept form.
y -3/2x 2
y 1/3x 5/3
Now graph and see where they intersect. Do they
cross at (2,-1) ?
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SOLVE

• -Graph and give solution then check (plug
  solution into each equation)

y x 1 y -x 5
Solution (2, 3)
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Solve If in standard form, rewrite in
slope-intercept form, graph the lines, then plug
in to check.
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• 2x y 4

y -2x 4
x y 2
y x - 2
Y
y x (-2)
X
2
-2
y -2x 4
Solution (2,0)
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Check

• 2x y 4 x y 2
• 2(2) 0 4 2 0 2
• 4 4 2 2
• Both equations work with the same solution, so
  (2,0) is the solution to the system.

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Example 1

• If you invest 9,000 at 5 and 6 interest, and
  you earn 510 in total interest, how much did you
  invest in each account?

Equation 1 .05x .06y 510
Equation 2 x y 9,000
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• Solve by graphing (find the x, y-intercepts)
• When x 0 When y 0

.05x 510 x 10,200
.06y 510 y 8,500
--------------------------------------------------
------------------------------
x y 9,000 y 9,000
x y 9,000 x 9,000
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Investment
(3,000, 6,000) Solution
1 2 3 4 5 6 7 8
9 10
Thousands at 6
1 2 3 4 5 6 7 8
9 10 11
Thousands at 5
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• Graph is upper right quadrant, crossing at
  (3,000, 6,000)

Answer 3,000 is invested at 5 and 6,000 is
invested at 6 CHECK ANSWER TO MAKE SURE!!