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Algebra

Algebra. Chapter 7. Vocabulary. System of linear equations- two or more linear equations in the same variables. Solution of a system of linear equations- an ordered pair (x, y) that satisfies each equation. . Re-teach. Solving a linear system using graph- and –check

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Algebra

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  1. Algebra Chapter 7

  2. Vocabulary System of linear equations- two or more linear equations in the same variables. Solution of a system of linear equations- an ordered pair (x, y) that satisfies each equation.

  3. Re-teach Solving a linear system using graph- and –check • Write the equation in a form so it is easy to graph. • Graph both equations. • Estimate the points (x,y) for intersection. • Check algebraically by substituting into each equation.

  4. Practice Graph the linear system, then decide if the ordered pair is a solution. • x + y = -2 (-3, 1) 2x – 3y = -9 2. –x + y = - 2 (4, -2) 2x + y = 10 3. x + 3y = 15 (3, -6) 4x + y = 6

  5. Re-teach Solving systems of equations by substitution • Solve one of the equations for one of its variables. • Substitute the revised Expression in for the other equation. • Solve the equation for the variable. • Substitute in the solution for one of the variables into the original equation. • Write the solution in an ordered pair.

  6. Re-teach Solving systems of equations by substitution -x + y = 1 2x + y = -2 • Solve one of the equations for one of its variables. -x + y = 1 + x + x Y = x + 1 2x + y = -2

  7. Re-teach Solving systems of equations by substitution 2) Substitute the revised Expression in for the other equation. Y = x + 1 2x + y = -2 2x + x + 1 = -2

  8. Re-teach Solving systems of equations by substitution 3) Solve the equation for the variable. 2x + x + 1 = -2 3x + 1 = -2 3x = -3 X = -1

  9. Re-teach Solving systems of equations by substitution 4) Substitute in the solution for one of the variables into the original equation. -x + y = 1 2x + y = -2 X = -1 2(-1) + y = -2 -2 + y = -2 Y = 0

  10. Re-teach Solving systems of equations by substitution 5) Write the solution in an ordered pair. X = -1 Y = 0 (-1, 0)

  11. Practice • 2x + 2y = 3 x – 4y = -1 2) –x + y = 5 ½x + y = 8 3) 3x + y = 3 7x + 2y = 1

  12. Vocabulary Linear Combination- an equation obtained by adding one of ht equations to the other equation. A linear combination is often known as solving systems of equations through elimination.

  13. Re-teach Solving linear systems by linear combinations • Arrange the equations with like terms in columns. • Multiply one or both of the equations to obtain an opposite variable. • Add/Subtract the terms from each column to get the value of the variable. • Substitute the value into the original equations. 5) Write the solution in an ordered pair.

  14. Re-teach Solving linear systems by linear combinations • Arrange the equations with like terms in columns. 3x + 2y = 44 5y + x = 11 3x + 2y = 44 X + 5y = 11

  15. Re-teach Solving linear systems by linear combinations 2) Multiply one or both of the equations to obtain an opposite variable. 3x + 2y = 44 -3(X + 5y = 11) 3x + 2y = 44 -3x -15y = -33

  16. Re-teach Solving linear systems by linear combinations 3) Add/Subtract then divide the terms from each column to get the value of one variable. 3x + 2y = 44 -3x -15y = -33 -11y = 11 Y = -1

  17. Re-teach Solving linear systems by linear combinations 4) Substitute the value into the original equation. Y = -1 3x + 2y = 44 5y + x = 11 5(-1) + x = 11 -5 + x = 11 X = 16

  18. Re-teach Solving linear systems by linear combinations 5) Write the solution as an ordered Pair. Y = -1 X = 16 (16, -1)

  19. Re-teach One solution (system will have perpendicular slopes) No solutions (system will have parallel slopes) Infinite solutions (system will be the same or equal 0 = 0)

  20. Practice Determine the type of results from each system. • 3x + y = -1 -9x – 3y = 3 2) X – 2y = 5 -2x + 4y = 2 3) 2x + y = 4 4x – 2y = 0

  21. Vocabulary System of linear inequalities- two or more linear inequalities. Solution- ordered pair of the inequality in each system. Graph of linear inequalities- graph of all solutions of the system.

  22. Warm Up When is a line dotted? When is a line solid? How do we determine which way to shade a graph?

  23. Re-teach Triangular Solution y < 2 x ≥ -1 y > x – 2 • Graph each system on the same plane. • The overlap is the intersection of the graphs. • After graphing pick a point, check to see if the point is a solution algebraically.

  24. Re-teach Solution between parallel lines Y < 3 Y > 1 • Graph the equations. • Determine the overlapping shaded area.

  25. Re-teach Quadrilateral Solution Region x ≥ 0 Y ≥ 0 Y ≤ 2 Y ≤ -½x + 3 • Graph the system of linear inequalities. • Label each intersecting point. • Shade the region inside the intersecting points.

  26. Practice Graph and determine the types of solutions from each system. • 2x + y < 4 -2x + y ≤ 4 2) 2x + y ≥ -4 x – 2y < 4 3) 2x + y ≤ 4 2x + y ≥ - 4

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