# Systems of Equations and Inequalities

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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 = x + 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 = – 5y = -16 solve equation y = -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 = x + 2y = 6 9x + 2y = (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 = x + 2y = 6 -1(9x + 2y = 22) x – 2y = -22 Now eliminate the 2y and -2y

Solving Systems of Equations by Elimination
Ex) 5x + 2y = 6 -9x – 2y = x = -16 x = 4 Now substitute the 4 in for x and solve for y 5(4) + 2y = y = 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 = y = 6 4y = 12 y = 3 solution (-2,3)

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

Solving Systems of Equations by Elimination
Ex) -3x – 3y = x + 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 = = -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 = = 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(1) < -3(5) < -3(1) + 3 5 < < < < 0 5 < < < < 1 + 2 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(1) < -3(5) < -3(1) + 3 5 < < < < 0 5 < < < < 1 + 2 5 < < < < 3

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