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Warm Up

Warm Up. Solve each with substitution. 2x+y = 6 y = -3x+5 3x+4y=4 y=-3x- 3. 4. What is the solution to a system of equations geometrically? 5.What are three types of solutions when solving systems of equations?.

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Warm Up

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  1. Warm Up Solve each with substitution. 2x+y = 6 y = -3x+5 3x+4y=4 y=-3x- 3 4. What is the solution to a system of equations geometrically?5.What are three types of solutions when solving systems of equations?

  2. Systems of Linear Equations: Elimination The Elimination method is used when both equations are in standard form and like terms have opposite coefficients. Ex: -4x + y = 9 -3x + 5y = -7 -2x – y = -1 3x + 2y = -4 both equations in standard form both equations in standard form y terms have opposite coefficients x terms have opposite coefficients Non Ex: y = 2x – 10 x – 6y = 1 5x – y = 9 x + 3y = 8 substitution method both equations in standard form No opposite coefficients (elimination using multiplication) Which is an example of a system of linear equations using the elimination method? x + 6y = -10 8x + 3y = -1 -x – y = 9 x = 3y – 2

  3. The Addition Property of equations- Two equations may be added left + left = right + right . Add the second equation to the first equation to eliminate the variables with opposite coefficients. x + y = 19 + x – y = -7 x – y and -7 are the same value added to both sides ____________________________ 2x = 12 x = 6 Substitute the value of x into either equation to solve for y What is the Addition Property? Why can you add the following system together? 3x – 5y = -4 x + 5y = 12

  4. Steps: 1. Add the equations to eliminate a variable. 2. Solve the remaining equation. 3. Substitute the solved variable into either equation. 4. Solve. 5. Write the solution. 6. Check.

  5. Ex 1: 2x – 3y = 0 -4x + 3y = -1 Steps: 1. Add the equations to eliminate a variable. 2. Solve the remaining equation. 3. Substitute the solved variable into either equation. 4. Solve. 5. Write the solution. 6. Check.

  6. Ex 2: x + 3y = 7 -x + 4y = 3 Steps: 1. Add the equations to eliminate a variable. 2. Solve the remaining equation. 3. Substitute the solved variable into either equation. 4. Solve. 5. Write the solution. 6. Check.

  7. Ex 3: 2x – 7y = -1 -2x + 7y = 1 Steps: 1. Add the equations to eliminate a variable. 2. Solve the remaining equation. 3. Substitute the solved variable into either equation. 4. Solve. 5. Write the solution. 6. Check.

  8. Ex 4: -3x – y = 12 3x + y = -11 Steps: 1. Add the equations to eliminate a variable. 2. Solve the remaining equation. 3. Substitute the solved variable into either equation. 4. Solve. 5. Write the solution. 6. Check.

  9. Ex 5: 5x – 6y = -32 3x + 6y = 48 Steps: 1. Add the equations to eliminate a variable. 2. Solve the remaining equation. 3. Substitute the solved variable into either equation. 4. Solve. 5. Write the solution. 6. Check.

  10. Ex 6: -5x + 2y = 8 5x + 2y = 8 Steps: 1. Add the equations to eliminate a variable. 2. Solve the remaining equation. 3. Substitute the solved variable into either equation. 4. Solve. 5. Write the solution. 6. Check.

  11. Ex 7: 2x + 4y = 17 6x – 4y = -9 Steps: 1. Add the equations to eliminate a variable. 2. Solve the remaining equation. 3. Substitute the solved variable into either equation. 4. Solve. 5. Write the solution. 6. Check.

  12. Ex 8: x + 7y = -14 -x – 7y = -2 Steps: 1. Add the equations to eliminate a variable. 2. Solve the remaining equation. 3. Substitute the solved variable into either equation. 4. Solve. 5. Write the solution. 6. Check.

  13. Systems of Linear Equations: Elimination using Multiplication The Elimination using Multiplication method is used when both equations are in standard form but like terms DO NOT have opposite coefficients. Ex: -x + 7y = 9 2x + 5y = -7 -2x – y = -1 3x + 5y = -4 both equations in standard form both equations in standard form Like terms do not have opposite coefficients Like terms do not have opposite coefficients Non Ex: y = 2x – 10 x – 6y = 1 5x – y = 9 3x + 6y = 8 substitution method both equations in standard form like terms have opposite coefficients Give an example of the Elimination using Multiplication method.

  14. Multiply one or both equations by a nonzero number to create opposite coefficients. Add the equations to eliminate one of the variables. • Is any coefficient a divisor of another? If so, multiply the equation with that divisor to create opposite coefficients 2x – 4y = -4 3x + 2y = -6 2 is a DIVISOR of 4 so we multiply the second equation to create opposite coefficients. How do you create opposite coefficients for a variable?

  15. Steps: 1. Find the lcm of two coefficients. 2. Multiply one or both equations to create opposite coefficients. 3. Add the equations. 4. Solve the remaining equation for one variable. 5. Substitute to find the other variable and write the solution. 6. Check.

  16. Ex 1: -14x + 6y = -9 -7x + 3y = 8 Steps: 1. Find the lcm of two coefficients. 2. Multiply one or both equations to create opposite coefficients. 3. Add the equations. 4. Solve the remaining equation for one variable. 5. Substitute to find the other variable and write the solution. 6. Check.

  17. Ex 2: 4x + 2y = 14 7x – 3y = -8 Steps: 1. Find the lcm of two coefficients. 2. Multiply one or both equations to create opposite coefficients. 3. Add the equations. 4. Solve the remaining equation for one variable. 5. Substitute to find the other variable and write the solution. 6. Check.

  18. Ex 3: 3x + 2y = -9 -10x + 5y = -5 Steps: 1. Find the lcm of two coefficients. 2. Multiply one or both equations to create opposite coefficients. 3. Add the equations. 4. Solve the remaining equation for one variable. 5. Substitute to find the other variable and write the solution. 6. Check.

  19. Ex 4: -4x – y = 7 20x + 5y = -35 Steps: 1. Find the lcm of two coefficients. 2. Multiply one or both equations to create opposite coefficients. 3. Add the equations. 4. Solve the remaining equation for one variable. 5. Substitute to find the other variable and write the solution. 6. Check.

  20. Ex 5: 3x – 2y = 10 2x + 3y = -2 Steps: 1. Find the lcm of two coefficients. 2. Multiply one or both equations to create opposite coefficients. 3. Add the equations. 4. Solve the remaining equation for one variable. 5. Substitute to find the other variable and write the solution. 6. Check.

  21. Ex 6: -3x + 15y = 12 -x + 5y = -4 Steps: 1. Find the lcm of two coefficients. 2. Multiply one or both equations to create opposite coefficients. 3. Add the equations. 4. Solve the remaining equation for one variable. 5. Substitute to find the other variable and write the solution. 6. Check.

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