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Systems of Linear Equations!

Systems of Linear Equations!. By graphing. Definition. A system of linear equations, aka linear system, consists of two or more linear equations with the same variables. x + 2y = 7 3x – 2y = 5. The solution.

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Systems of Linear Equations!

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  1. Systems of Linear Equations! By graphing

  2. Definition • A system of linear equations, aka linear system, consists of two or more linear equations with the same variables. • x + 2y = 7 • 3x – 2y = 5

  3. The solution • The solution of a system of linear equations is the ordered pair that satisfies each equation in the system. • One way to find the solution is by graphing. • The intersection of the graphs is the solution.

  4. Example X + 2y = 7 3x – 2y = 5 • Step 1: graph both equations • Step 2: estimate coordinates of the intersection • Step 3: check algebraically by subsitution

  5. Types of systems Consistent Independent System – has exactly one solution *other types to be discussed later

  6. More examples -5x + y = 0 5x + y = 10 -x + 2y = 3 2x + y = 4

  7. Multi-step problem x + y = 25 15x + 30y = 450 A business rents in line skates ad bicycles. During one day the businesses has a total of 25 rentals and collects $450 for the rentals. Find the total number of pairs of skates rented and the number of bicycles rented. Skates - $15 per day Bikes - $30 per day

  8. Now find the totals when there were only 20 rentals and they made $420.

  9. Solve by Substitution

  10. Steps 3x – y = -2 X + 2y = 11 3x + 2 = y X + 2(3x + 2) = 11 Step 2: substitute the expression in the other equation for the variable and solve X + 6x + 4 = 11 7x = 7 X = 1 Step 3: substitute the solution back into the equation from step 1 and solve 3(1) + 2 = y 5 = y Solution: (1,5) Step 1: Solve one of the equations for a variable

  11. More examples X – 2y = -6 4x + 6y = 4 Y = 2x + 5 3x + y = 10 3x + y = -7 -2x + 4y = 0

  12. Multi-step problem X + y = 26 15x + 7.5y = 360 A group of friends takes a day-long tubing trip down a river. The company that offers the tubing trip charges $15 to rent a tube for a person to use and $7.50 to rent a tube to carry the food and water in a cooler. The friends spend $360 to rent a total of 26 tubes. How many of each type of tube do they rent?

  13. Elimination 7.3

  14. Elimination Method 2x + 3y = 11 -2x + 5y = 13 8y = 24 (1,3) Step 2: Solve the resulting equation for the other variable. 8y = 24 Y = 3 Step 3: Substitute into either original equation to find the value of the other variable. 2x + 3(3) = 11 2x + 9 = 11 2x = 2 X = 1 Step 1: Add the equations to eliminate one variable.

  15. A little twist Step P: Make Opposite Step 1: Add Step 2: Solve Step 3: Substitute/Solve 4x + 3y = 2 5x + 3y = -2 4x + 3y = 2 -5x – 3y = 2 -1( ) -x = 4 (-4, 6) X = -4 4(-4) + 3y = 2 -16 + 3y = 2 3y = 18 Y = 6

  16. Arranging like terms If two linear systems are not in the same form you must rearrange one! 8x – 4y = -4 4y = 3x + 14

  17. Examples You try: 4x – 3y = 5 -2x + 3y = -7 -5x – 6y = 8 5x + 2y = 4 3x + 4y = -6 2y = 3x + 6 7x – 2y = 5 7x – 3y = 4 2x + 5y = 12 5y = 4x + 6

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