Systems of Equations and Inequalities Algebra I

Vocabulary System of (linear) equations Two equations together. An ordered pair that satisfies both equations (where they cross on the graph) Can have 0, 1 or infinite number of solutions.

Vocabulary Intersecting graph – Two lines that intersect or coincide – called consistent. Parallel graph – Two lines that are parallel to each other – called inconsistent. Same line graph – Two lines that graph on top of each other exactly. Independent system – A system that has exactly one solution.

Intersecting Lines Exactly one solution – the point where the two lines intersect is the solution. Consistent and independent.

Parallel Lines No solutions. Inconsistent

Same Line Infinite solutions – they intersect at every point. Consistent and dependent.

Graph on the calculator Equations must always be in slope-intercept form (y = mx + b) Enter into the y= function in the calculator Graph

Example y = -x + 5 y = x – 3

Example y = -x + 5 y = x – 3 One solution

Now You Try… 1. y = -x + 5 2x + 2y = -8 2. 2x + 2y = -8 y = -x - 4

Now You Try… 1. y = -x + 5 2x + 2y = -8 (y = -x – 4) No solutions 2. 2x + 2y = -8 (y = -x – 4) y = -x – 4 Infinite solutions

Solving Systems of Equations The exact solution of a system of equation can be found using algebraic methods. Can solve by: Substitution Elimination Graphing

Solving Systems of Equations by Substitution Ex) y = 3x x + 2y = -21

Solving Systems of Equations by Substitution Ex) y = 3x x + 2y = -21 Since we already know that y = 3x, substitute 3x into the second equation and solve for x. x + 2(3x) = -21

Solving Systems of Equations by Substitution Ex) x + 2(3x) = -21 x + 6x = -21 distribute 7x = -21 combine like terms x = -3 divide by 7 Now substitute the value for x into the first equation to solve for y. y = 3(-3)

Solving Systems of Equations by Substitution We now know that… x = -3 and y = -9 The solution is (-3,-9) This is the point where the two lines intersect on the graph.

Solving Systems of Equations by Substitution Sometimes, you need to get one variable by itself to use substitution method. x + 5y = -3 3x – 2y = 8

Solving Systems of Equations by Substitution Sometimes, you need to get one variable by itself to use substitution method. x + 5y = -3 3x – 2y = 8 - 5y -5y x = -5y – 3 Now substitute into the second equation and solve. 3(-5y – 3) – 2y = 8

Solving Systems of Equations by Substitution 3(-5y – 3) – 2y = 8 -15y – 9 – 2y = 8 distribute -17y – 9 = 8 combine like terms -17y = 17 add 9 to both sides y = -1 divide by -17 Substitute -1 into the first equation for y and solve for x. x = -5y – 3 x = -5(-1) – 3 = 2 solution (2, -1)

Solving Systems of Equations by Elimination Solving by elimination can be done by addition or multiplication.

Solving Systems of Equations by Elimination Solve by addition Ex) 3x – 5y = -16 2x + 5y = 31 Notice that there is an inverse here (-5y and 5y)

Solving Systems of Equations by Elimination 3x – 5y = -16 the -5y and 5y will cancel +2x + 5y = 31 add like terms 5x + 0 = 15 divide by 5 x = 3 Now substitute the 3 into either equation for x and solve for y. 3(3) – 5y = -16 9 – 5y = -16 solve equation -9 -9 -5y = -25 y = 5 solution (3,5)

Solving Systems of Equations by Elimination Solve by multiplication If there is not an inverse, you need to multiply one or both of the equations to make an inverse in the problem. Ex) 5x + 2y = 6 9x + 2y = 22

Solving Systems of Equations by Elimination Solve by multiplication If there is not an inverse, you need to multiply one or both of the equations to make an inverse in the problem. Ex) 5x + 2y = 6 5x + 2y = 6 9x + 2y = 22 -1(9x + 2y = 22) Now eliminate the 2y and -2y

Solving Systems of Equations by Elimination Solve by multiplication If there is not an inverse, you need to multiply one or both of the equations to make an inverse in the problem. Ex) 5x + 2y = 6 5x + 2y = 6 -1(9x + 2y = 22) -9x – 2y = -22 Now eliminate the 2y and -2y

Solving Systems of Equations by Elimination Ex) 5x + 2y = 6 -9x – 2y = -22 -4x = -16 x = 4 Now substitute the 4 in for x and solve for y 5(4) + 2y = 6 20 + 2y = 6 2y = -14 y = -7 solution (4, -7)

Solving Systems of Equations by Elimination Ex) 3x + 4y = 6 5x + 2y = -4

Solving Systems of Equations by Elimination Ex) 3x + 4y = 6 5x + 2y = -4 Sometimes, there is nothing obvious to inverse, you may have to multiply one or both equations by a number to inverse. 3x + 4y = 6 3x + 4y = 6 -2(5x + 2y = 4) -10x – 4y = 8

Solving Systems of Equations by Elimination 3x + 4y = 6 -10x – 4y = 8 -7x = 14 x = -2 3(-2) + 4y = 6 -6 + 4y = 6 4y = 12 y = 3 solution (-2,3)

Solving Systems of Equations by Elimination Ex) -3x – 3y = -21 -2x + 8y = 16

Solving Systems of Equations by Elimination Ex) -3x – 3y = -21 -2x + 8y = 16 When neither equation has anything in common, you will have to multiply BOTH equations to find an inverse. -2(-3x – 3y = -21) 6x + 6y = 42 3(-2x + 8y = 16) -6x + 24y = 48

Solving Systems of Equations by Elimination 6x + 6y = 42 -6x + 24y = 48 30y = 90 y = 3 6x + 6(3) = 42 6x + 18 = 42 6x = 24 x = 4 solution (4, 3)

Solving Systems of Equations Ex) 3x – 6y = 10 x – 2y = 4

Solving Systems of Equations with No Solution Ex) 3x – 6y = 10 x – 2y = 4 Make inverse 3x – 6y = 10 3x – 6y = 10 -3(x – 2y = 4) -3x + 6y = -12 0 = -2 0 ≠ -2 therefore, there is no solution

Solving Systems of Equations Ex) 3x + 6y = 24 -2x – 4y = -16

Solving Systems of Equations with Infinite Solutions Ex) 3x + 6y = 24 -2x – 4y = -16 Find the inverse 2(3x + 6y = 24) 6x + 12y = 48 3(-2x – 4y = -16) -6x – 12y = -48 0 = 0 0 = 0 is a true statement, there are infinite solutions. They would graph as the same line.

Solving Systems of Equations Method Best time to use Graphing *to estimate a solution Substitution *when one variable has a coefficient of 1 or -1 Elimination *when one of the variables has the same or opposite coefficients. *when there are no other options for solving.

Solving Systems of Inequalities Vocabulary < Less than symbol (dotted line) > Greater than symbol (dotted line)

Solving Systems of Inequalities Vocabulary < Less than or equal to symbol (solid line) > Greater than or equal to symbol (solid line)

Solving Systems of Inequalities Solve these by graphing: Change inequality into slope intercept form. Put inequality into the y= function on your calculator. Graph and shade depending on the sign. Solution is the area on the coordinate plane that is shaded by both inequalities.

Solving Systems of Inequalities Solution to x < 1 and y < 3

Solving Systems of Inequalities by Graphing

Solving Systems of Inequalities To determine if a point is in the solution set of a system of inequalities. Substitute the value in to the inequality If it is a true statement, they are part of the solution set, if not, they aren’t. Ex) y < -3x + 3 y < x + 2 (-1, 5), ( 1, 5), (5, 1), (1, -5)

Solving Systems of Inequalities Ex) y < -3x + 3 y < x + 2 (-1, 5), ( 1, 5), (5, 1), (1, -5) 5 < -3(-1) + 3 5 < -3(1) + 3 1 < -3(5) + 3 -5 < -3(1) + 3 5 < 6 5 < 0 1 < -12 -5 < 0 5 < -1 + 2 5 < 1 + 2 1 < 5 + 2 -5 < 1 + 2 5 < 1 5 < 3 1 < 7 -5 < 3

Solving Systems of Inequalities Ex) y < -3x + 3 y < x + 2 (-1, 5), ( 1, 5), (5, 1), (1, -5) 5 < -3(-1) + 3 5 < -3(1) + 3 1 < -3(5) + 3 -5 < -3(1) + 3 5 < 6 5 < 0 1 < -12 -5 < 0 5 < -1 + 2 5 < 1 + 2 1 < 5 + 2 -5 < 1 + 2 5 < 1 5 < 3 1 < 7 -5 < 3