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Understanding and Solving Linear Systems by Graphing

This guide explores the concept of a system of two linear equations in two variables, with emphasis on graphing solutions. It includes examples that demonstrate how to determine if ordered pairs are solutions by substitution and graphing. Key graphical features, such as identifying the point of intersection, which represents the solution, are discussed alongside methods for checking solutions algebraically. It also covers cases where systems have one solution, no solution, or infinitely many solutions, aiding in a comprehensive understanding of linear systems.

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Understanding and Solving Linear Systems by Graphing

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  1. 3.1 Solving Linear Systems by Graphing 9/20/13

  2. Vocabulary System of 2 Linear Equations: A system consisting of two linear equations in two variables. Ex: 6x – 2y = 8 3x – y = 4 Solution of a system of 2 linear equations: Is an ordered pair (x, y) that satisfies both equations. Graphically, it’s the point where the lines intersect.

  3. Tell whether the ordered pair (3, 4) is a solution of -2x + y = -2 4x – 2y = 3 Substitute 3 for x and 4 for y in BOTH equations. -2(3) + 4 = -2 - 6 + 4 = -2 4(3) – 2(4) = 3 12 – 8 = 3 Answer: Not a Solution

  4. Tell whether the ordered pair (3, 4) is a solution of x + 2y = 11 2x – y = 2 Substitute 3 for x and 4 for y in BOTH equations. 3 + 2(4) = 11 3 + 8 = 11 2(3) – 4 = 2 6 – 4 = 2 Answer: Solution

  5. + = y 2 x 9 + = y – x 3 ANSWER – ( 2, 5 ) Example 1 Solve a System by Graphing Solve the system by graphing. Then check your solution.

  6. You can check the solution by substituting -2for x and 5for y into the original equations. y= - x + 3 5= -(-2) + 3 5= 5 y = 2 x + 9 5 = 2(-2) + 9 5 = -4 + 9 5 = 5

  7. Standard  Slope Int Form • Add or subtract the x – term on both sides of the equation. • Divide everything by the coefficient of y if the coefficient is not 1. Ex. Ex.

  8. Example 2 = 3x – y 3 = x + 2y 8 ANSWER ( 2,3 ) Solve a System by Graphing Solve the system by graphing. Then check your solution algebraically. In slope int. form: y = 3x - 3 In slope int. form:y = - x + 4

  9. Example 2 ? ? = = Equation 1 Equation 2 ( 2, 3 ). = x + 8 2y ? 2 8 2 + 3 – 3 3 = The solution of the system is ? 6 – 3 3 6 = 2 8 + = 3x – y 3 = = 3 8 3 8 ANSWER ( ( ) ) 2 3 Solve a System by Graphing You can check the solution by substituting 2 for x and 3 for y into the original equations.

  10. Extra Example – – + – = x y 1 = x 3y 1 ANSWER ( 1, 0 ) Solve the system by graphing. Then check your solution. 2.

  11. Checkpoint ANSWER ( 2, 1 ) Solve a System by Graphing Solve the system by graphing. Then check your solution.

  12. Homework WS 3.1. Do all work on the worksheet. Pencil only. Use straight edge/Ruler

  13. Number of Solutions 1 solution : the lines have different slopes No solution :the lines are parallel (same slope) Infinitely many solutions :the lines have the same equation.

  14. b. – 2x y 1 = x + 2y 4 = – – 4x + 2y 2 = x + 2y 1 = Example 3 Systems with Many or No Solutions Tell how many solutions the linear system has. a. Infinitely many solutions :the lines have the same equation. No solution :the lines are parallel (same slope)

  15. Checkpoint 3. 1. 2x + 3y 1 = 0 ANSWER 4x + 6y 3 = 1 ANSWER 2. – x 4y 5 = – – x + 4y 5 = infinitely many solutions ANSWER – x 5y 5 = x + 5y 5 = Write and Use Linear Systems Tell how many solutions the linear system has without graphing.

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