Chapter 8 Section 1 Solving System of Equations Graphically

Learning Objective Determine if the ordered pair is a solution to a system of equations. Determine if a system of equations is consistent, inconsistent or dependent. Solve a system of equations graphically

Key Vocabulary system of linear equations solution to a system of equations consistent inconsistent dependent independent

Solution to a System of Equations 1.A System of Linear Equations is when we seek a common solution to two or more linear equation. y = x + 5 y = 2x The solutions to a system of equation are ordered pair or pairs that satisfy all equations in the system.

Determine if an Ordered Pair is a Solution to a System of Equations Example: (2, 4) Equation # 1Equation # 2 y = 3x – 25x – 2y = 2 4 = 3(2) – 25(2) – 2(4) = 2 4 = 6 – 210 – 8 = 2 4 = 42 = 2True Yes, is a solution

Determine if an Ordered Pair is a Solution to a System of Equations Example: (3, 3) Equation # 1Equation # 2 y = 3x – 25x – 2y = 2 3 = 3(3) – 25(3) – 2(3) = 2 3 = 9 – 215 – 6 = 2 3 ≠ 7 9 ≠ 2False No, not a solution

Determine if an Ordered Pair is a Solution to a System of Equations Example: (3, 1) Equation # 1Equation # 2 y = 3x – 84x – 3y = 9 1 = 3(3) – 84(3) – 3(1) = 9 1 = 9 – 812 – 3 = 9 1 = 19 = 9True Yes, not a solution

Determine if an Ordered Pair is a Solution to a System of Equations Example: (5, 7) Equation # 1Equation # 2 y = 3x – 84x – 3y = 9 7 = 3(5) – 84(5) – 3(7) = 9 7 = 15 – 820 – 21 = 9 7 = 71 ≠ 9 TrueFalse No, not a solution

Solution to a System of Equations There are three methods of finding a solution to a system of equations.  Graphically – we will learn this method in this section (8.1)  Substitution – we will learn in section 8.2  Addition – we will learn in section 8.3

Solution to a System of Equations Graphically – are ordered pairs that are common to all lines when graphed. 3 situation are possible. Consistent system – two nonparallel lines. Has exactly one solution. Inconsistent system – two parallel line. Does not have a solution. Lines do not intersect. Dependent system – Same line. Has infinite number of solution, therefore, every point on the line satisfies both equation. Line 2 Line 1 Line 1 and 2

Solution to a System of Equations To determine if consistent, inconsistent, or dependent write the equations in slope intercept form (y = mx + b) and compare the slopes and y-intercepts.  Consistent – One solution, different slopes, different y-intercept  Inconsistent – Parallel lines, no Solution, same slope, different y-intercept  Dependent – Infinite number of solutions, same slope, same y-intercept

Determine if the systems have exactly one solution, no solution, or an infinite number of solutions. Identify if consistent, inconsistent, or dependent Example: Equation # 1Equation # 2 Different slope, Different y-intercept Consistent One solution

Determine if the systems have exactly one solution, no solution, or an infinite number of solutions. Identify if consistent, inconsistent, or dependent Example: Equation # 1Equation # 2 Same slope, Different y-intercept Parallel Lines Inconsistent No Solution

Determine if the systems have exactly one solution, no solution, or an infinite number of solutions. Identify if consistent, inconsistent, or dependent Example: Equation # 1Equation # 2 Same slopes, same y-intercept Dependent Infinite number of solutions

Solve the system of equations graphically Example: y = -x + 4 2x + 2y = 0 2y = -2x y = -xSlope = -1 y-int = (0, 4)y-int = (0,0)Negative slopeDown oneRight one (0,4) (0,0) y = -x + 4 2x + 2y = 0 Same slopes, different y-intercepts Inconsistent System Two parallel line. Does not have a solution.

Solve the system of equations graphically Example: x + y = -1 y = 2x - 7 y = -x – 1 Slope = -1Slope = 2 y-int = (0, -1)y-int = (0,-7) Negative slopePositive slope Down oneUp twoRight one (2,-3) (0,0) y = 2x - 7 x + y = -1 Different slope, different y-intercept Consistent System One solution (2, -3)

Solve the system of equations graphically Example: y = ½ x + 1 2y – x = 2 2y = x + 2 y = ½ x + 1 Slope = ½ y-int = (0,1)Positive slopeUp oneRight two (0,1) (0,0) y = ½ x + 1 2y –x = 20 Same slope, same y-intercept Dependent system Same line Infinite number of solution Every point on the line satisfies both equation.

Remember The solution to a system of equations is an ordered pair. Check by substituting the ordered pair back into the original equation. Be careful when you graph using fractions because it is not always easy to read.

HOMEWORK 8.1 Page : #9, 13, 17, 18, 19, 37, 43, 45, 57