systems of equations and inequalities algebra i. vocabulary system of (linear) equations two...
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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.
Graph on the calculator
• Equations must always be in slope-intercept form (y = mx + b)
• Enter into the y= function in the calculator
• Graph
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
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) = -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)
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 = -33x – 2y = 8
Solving Systems of Equations by Substitution
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
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 -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.
Solving Systems of Equations by Elimination
• Solve by additionEx)3x – 5y = -162x + 5y = 31Notice 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 = 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)
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 = 4Now substitute the 4 in for x and solve for y5(4) + 2y = 620 + 2y = 6 2y = -14 y = -7 solution (4, -7)
Solving Systems of Equations by Elimination
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
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 = 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
Solving Systems of Equations by Elimination
6x + 6y = 42-6x + 24y = 48 30y = 90 y = 36x + 6(3) = 426x + 18 = 42 6x = 24 x = 4 solution (4, 3)
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
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.
Solving Systems of Equations
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)
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
• 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) + 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