Systems of Equations and Inequalities Algebra I. Vocabulary System of (linear) equations Two equations together. An ordered pair that satisfies both equations.

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

1. y = -x + 5 y = x – 3One 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 solutions2. 2x + 2y = -8 (y = -x – 4) y = -x – 4Infinite 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


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


Ex) x + 2(3x) = -21x + 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)


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.


Sometimes, you need to get one variable by itself to use substitution method.x + 5y = -33x – 2y = 8


Sometimes, you need to get one variable by itself to use substitution method.x + 5y = -33x – 2y = 8

x + 5y = -3 - 5y -5yx = -5y – 3Now substitute into the second equation and solve.3(-5y – 3) – 2y = 8


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 -17Substitute -1 into the first equation for y and solve for x.x = -5y – 3x = -5(-1) – 3 = 2 solution (2, -1)

Solving Systems of Equations by Elimination

• Solving by elimination can be done by addition or multiplication.


• Solve by additionEx)3x – 5y = -162x + 5y = 31Notice that there is an inverse here (-5y and 5y)


3x – 5y = -16 the -5y and 5y will cancel+2x + 5y = 31 add like terms 5x + 0 = 15 divide by 5 x = 3Now substitute the 3 into either equation for x and solve for y.3(3) – 5y = -169 – 5y = -16 solve equation-9 -9-5y = -25 y = 5 solution (3,5)


• 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




Ex) 5x + 2y = 6 5x + 2y = 6 9x + 2y = 22 -1(9x + 2y = 22)• Now eliminate the 2y and -2y




Ex) 5x + 2y = 6 5x + 2y = 6-1(9x + 2y = 22) -9x – 2y = -22• Now eliminate the 2y and -2y


Ex) 5x + 2y = 6-9x – 2y = -22 -4x = -16 x = 4Now substitute the 4 in for x and solve for y5(4) + 2y = 620 + 2y = 6 2y = -14 y = -7 solution (4, -7)


Ex) 3x + 4y = 6 5x + 2y = -4


Ex) 3x + 4y = 6 5x + 2y = -4Sometimes, 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


3x + 4y = 6 -10x – 4y = 8 -7x = 14 x = -2 3(-2) + 4y = 6 -6 + 4y = 6 4y = 12 y = 3 solution (-2,3)


Ex) -3x – 3y = -21 -2x + 8y = 16


Ex) -3x – 3y = -21 -2x + 8y = 16When 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


6x + 6y = 42-6x + 24y = 48 30y = 90 y = 36x + 6(3) = 426x + 18 = 42 6x = 24 x = 4 solution (4, 3)


Ex) 3x – 6y = 10 x – 2y = 4

Solving Systems of Equations with No Solution

Ex) 3x – 6y = 10 x – 2y = 4Make inverse 3x – 6y = 10 3x – 6y = 10 -3(x – 2y = 4) -3x + 6y = -12 0 = -2

0 ≠ -2 therefore, there is no solution


Ex) 3x + 6y = 24 -2x – 4y = -16

Solving Systems of Equations with Infinite Solutions

Ex) 3x + 6y = 24 -2x – 4y = -16Find the inverse 2(3x + 6y = 24) 6x + 12y = 483(-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.


Method Best time to useGraphing *to estimate a solutionSubstitution *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)


Vocabulary < Less than or equal to symbol (solid line)

> Greater than or equal to symbol (solid line)


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.


Solution to x < 1 and y < 3

Solving Systems of Inequalities by Graphing


• 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)


• 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) + 35 < 6 5 < 0 1 < -12 -5 < 05 < -1 + 2 5 < 1 + 2 1 < 5 + 2 -5 < 1 + 25 < 1 5 < 3 1 < 7 -5 < 3


• 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) + 35 < 6 5 < 0 1 < -12 -5 < 05 < -1 + 2 5 < 1 + 2 1 < 5 + 2 -5 < 1 + 25 < 1 5 < 3 1 < 7 -5 < 3

Systems of Equations and Inequalities Algebra I. Vocabulary System of (linear) equations Two equations together. An ordered pair that satisfies both equations.

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