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Section 7.1 “Solve Linear Systems by Graphing”

Section 7.1 “Solve Linear Systems by Graphing”. Section 7.2 “Solve Linear Systems by Substitution”. “Solve Linear Systems by Substituting”. y = 3x + 2. Equation 1. Equation 2. x + 2y = 11. x + 2y = 11. x + 2 (3x + 2) = 11. Substitute. x + 6x + 4 = 11. 7x + 4 = 11. x = 1.

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Section 7.1 “Solve Linear Systems by Graphing”

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  1. Section 7.1“Solve Linear Systems by Graphing”

  2. Section 7.2“Solve Linear Systems by Substitution”

  3. “Solve Linear Systems by Substituting” y = 3x + 2 Equation 1 Equation 2 x + 2y = 11 x + 2y = 11 x + 2(3x + 2) = 11 Substitute x + 6x + 4 = 11 7x + 4 = 11 x = 1 y = 3x + 2 Equation 1 Substitute value for x into the original equation y = 3(1) + 2 y = 5 (5) = 3(1) + 2 5 = 5 (1) + 2(5) = 11 11 = 11 The solution is the point (1,5). Substitute (1,5) into both equations to check.

  4. Section 7.3 “Solve Linear Systems by Adding or Subtracting” • ELIMINATION- adding or subtracting equations to obtain a new equation in one variable.

  5. “Solve Linear Systems by Elimination Multiplying First!!” Multiply First Eliminated x (2) 4x + 5y = 35 8x + 10y = 70 Equation 1 + x (-5) 15x - 10y = 45 -3x + 2y = -9 Equation 2 23x = 115 x = 5 4x + 5y = 35 Equation 1 Substitute value for x into either of the original equations 4(5) + 5y = 35 20 + 5y = 35 y = 3 4(5) + 5(3) = 35 35 = 35 -3(5) + 2(3) = -9 -9 = -9 The solution is the point (5,3). Substitute (5,3) into both equations to check.

  6. Section 7.5 “Solve Special Types of Linear Systems” LINEAR SYSTEM- consists of two or more linear equations in the same variables. Types of solutions: (1) a single point of intersection – intersecting lines (2) no solution – parallel lines (3) infinitely many solutions– when two equations represent the same line

  7. “Solve Linear Systems with No Solution” Eliminated Eliminated 3x + 2y = 10 Equation 1 _ + -3x + (-2y) = -2 3x + 2y = 2 Equation 2 This is a false statement, therefore the system has no solution. 0 = 8 “Inconsistent System” No Solution By looking at the graph, the lines are PARALLEL and therefore will never intersect.

  8. “Solve Linear Systems with Infinitely Many Solutions” x – 2y = -4 Equation 1 y = ½x + 2 Use ‘Substitution’ because we know what y is equals. Equation 2 Equation 1 x – 2y = -4 x – 2(½x + 2) = -4 x – x – 4 = -4 This is a true statement, therefore the system has infinitely many solutions. -4 = -4 “Consistent Dependent System” Infinitely Many Solutions By looking at the graph, the lines are the SAME and therefore intersect at every point, INFINITELY!

  9. Section 7.6 “Solve Systems of Linear Inequalities” SYSTEM OF INEQUALITIES- consists of two or more linear inequalities in the same variables. x – y > 7 Inequality 1 2x + y < 8 Inequality 2 A solution to a system of inequalities is an ordered pair (a point) that is a solution to both linear inequalities.

  10. ? ? 0 < 5 – 4 0≥-5 + 3 0 < 1 0≥ -2 Graph a System of Inequalities y < x – 4 Inequality 1 y ≥ -x + 3 Inequality 2 Graph both inequalities in the same coordinate plane. The graph of the system is the intersection of the two half-planes, which is shown as the darker shade of blue. (5,0)

  11. ? ? ? x + 2y ≤ 4 y ≥ -1 x > -2 0≥ -1 0 + 0 ≤ 4 0 > -2 Graph a System of THREE Inequalities y ≥ -1 Inequality 1: Graph all three inequalities in the same coordinate plane. The graph of the system is the triangular region, which is shown as the darker shade of blue. x > -2 Inequality 2: x + 2y ≤ 4 Inequality 3: Check (0,0)

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