Transcript
Solving Linear Solving Linear Equations Equations
AndAndInequalitiesInequalities
By: Raza ZaidiBy: Raza ZaidiAndAnd
Ahmed Ammar AwanAhmed Ammar Awan
Combining Like TermsCombining Like Terms
Like terms are terms that have Like terms are terms that have the same variables with the same the same variables with the same exponent.exponent.
Combining Like Terms is a Combining Like Terms is a process used to simplify an process used to simplify an expression or an equation using expression or an equation using addition and subtraction of the addition and subtraction of the coefficients of terms .coefficients of terms .
Combining Like TermsCombining Like Terms
For example there is a term:For example there is a term:
+2 - 5-6 3x x
−4x −2
Combining Like TermsCombining Like Terms
Q) Solve:Q) Solve:
5x² + 7x + 2 - 2x² + 7 + x²5x² + 7x + 2 - 2x² + 7 + x²
Solution:Solution:4x²+7x+2+74x²+7x+2+7
4x²+7x+94x²+7x+9
Combining Like TermsCombining Like Terms
Q) Solve:Q) Solve:2(x-y)+2x+32(x-y)+2x+3
Solution:Solution:
2x-2y+2x+32x-2y+2x+3
4x-2y+3 4x-2y+3
The Addition PropertyThe Addition Propertyofof
EqualityEquality
The Addition Property of The Addition Property of EqualityEquality
Linear Equation in One VariableLinear Equation in One Variable
An equation that can be written An equation that can be written in the formin the form: :
ax + b = cax + b = c
where a, b, and c are constants.where a, b, and c are constants.
The Addition Property of The Addition Property of EqualityEquality
Check Solutions to EquationsCheck Solutions to Equations
Determine if the number following Determine if the number following the equation is a solution to the the equation is a solution to the
equation.equation.
2x-3=5,42x-3=5,4
The Addition Property of The Addition Property of EqualityEquality
Solution:Solution:2x-3=52x-3=5Put x=4Put x=4
2(4)-3=52(4)-3=58-3=58-3=5
5=5 5=5 (True)(True)
The Addition Property of The Addition Property of EqualityEquality
Q) Solve equation and check your Q) Solve equation and check your solution.solution.
x-16=36x-16=36Solution:Solution:
Add 16 on both sidesAdd 16 on both sidesx-16+16=36+16x-16+16=36+16
x=52x=52Check:Check:
Put x=52 in the given equationPut x=52 in the given equation52-16=3652-16=36
36=3636=36 (True)(True)
The Multiplication PropertyThe Multiplication Propertyofof
EqualityEquality
The Multiplication Property of The Multiplication Property of EqualityEquality
In this section we will solve In this section we will solve equation of the form of:equation of the form of:
ax=bax=bBy using the multiplication By using the multiplication
propertypropertyof equality.of equality.
The Multiplication Property of The Multiplication Property of EqualityEquality
Q) Q) Solve:Solve:x+x+x=15x+x+x=15
Solution:Solution:3x=153x=15
Multiplying 1/3 on Both SidesMultiplying 1/3 on Both Sides
3x/3=15/33x/3=15/3
x=5x=5
The Multiplication Property of The Multiplication Property of EqualityEquality
Q) Q) Solve:Solve:3x/5=63x/5=6
Solution:Solution:Multiply 5 on both sidesMultiply 5 on both sides
5*3x/5=6*55*3x/5=6*53x=303x=30
Divide 3 on both sidesDivide 3 on both sides3x/3=30/33x/3=30/3
x=10x=10
Solving Linear EquationsSolving Linear Equationswith a Variable onwith a Variable on
Only One SideOnly One SideOf The EquationOf The Equation
Solving Linear Equations With a Variable Solving Linear Equations With a Variable on Only One Side Of The Equationon Only One Side Of The Equation
Now we will discuss how to solve Now we will discuss how to solve linear equations using both the linear equations using both the
addition and multiplication addition and multiplication properties of equality when a properties of equality when a
variable appears on only one side variable appears on only one side of equationof equation
Solving Linear Equations With a Variable Solving Linear Equations With a Variable on Only One Side Of The Equationon Only One Side Of The Equation
Q) Q) Solve:Solve:10-3x=710-3x=7
Solution:Solution:Subtract 10 from both sidesSubtract 10 from both sides
10-3x-10=7-1010-3x-10=7-10-3x=-3-3x=-3
Multiply -1/3 on both sidesMultiply -1/3 on both sides
-3x/-3=-3/-3-3x/-3=-3/-3x=1x=1
Solving Linear Equations With a Variable Solving Linear Equations With a Variable on Only One Side Of The Equationon Only One Side Of The Equation
Q) Q) Solve:Solve:3(x - 2)=123(x - 2)=12
Solution:Solution:3x-6=123x-6=12
Add 6 on both sidesAdd 6 on both sides
3x-6+6=12+63x-6+6=12+63x=183x=18
Multiply 1/3 on both sidesMultiply 1/3 on both sides
3x/3=18/33x/3=18/3x=6x=6
Solving Linear EquationsSolving Linear Equationswith the Variable onwith the Variable on
Both SideBoth SideOf The EquationOf The Equation
Solving Linear Equations with the Solving Linear Equations with the Variable on Both Side Of The EquationVariable on Both Side Of The Equation
Now we will discuss how to solve Now we will discuss how to solve linear equations using both the linear equations using both the
addition and multiplication addition and multiplication properties of equality when a properties of equality when a
variable appears on both side of variable appears on both side of equation.equation.
Solving Linear Equations with the Solving Linear Equations with the Variable on Both Side Of The EquationVariable on Both Side Of The Equation
Q) Q) Solve:Solve:4x+6=2x+44x+6=2x+4
Solution:Solution:Subtract 4x on both sidesSubtract 4x on both sides
4x+6-4x=2x+4-4x4x+6-4x=2x+4-4x6=4-2x6=4-2x
Subtract 4 on both sidesSubtract 4 on both sides6-4=4-2x-46-4=4-2x-4
2=-2x2=-2xMultiply -1/2 on both sidesMultiply -1/2 on both sides
2/-2 =-2x/-22/-2 =-2x/-2-1=x-1=x
Solving Linear Equations with the Solving Linear Equations with the Variable on Both Side Of The EquationVariable on Both Side Of The Equation
Q) Q) Solve:Solve:5x=2(x+6)5x=2(x+6)
Solutions:Solutions:5x=2x+125x=2x+12
Subtract 2x from both sides Subtract 2x from both sides 5x-2x=2x+12-2x5x-2x=2x+12-2x
3x=123x=12Multiply 1/3 on both sidesMultiply 1/3 on both sides
3x/3=12/33x/3=12/3x=4x=4
RatiosRatiosAndAnd
ProportionsProportions
Ratios And ProportionsRatios And Proportions
An equation in which two ratios are An equation in which two ratios are equal is called a equal is called a proportionproportion
A proportion can be written using A proportion can be written using colon notation like this colon notation like this a : b :: c : da : b :: c : d
or as the more recognizable (and or as the more recognizable (and useable) equivalence of two useable) equivalence of two fractions.fractions.__ a a__ = ___ = _ c c____
b db d
Ratios And ProportionsRatios And Proportions
Q) If you can buy one Toffee for 2 Rupee Q) If you can buy one Toffee for 2 Rupee then how many Toffees you buy with 10 then how many Toffees you buy with 10 Rupee?Rupee?
Solution:Solution:
1 toffee1 toffee = = x x 2 Rs 10 Rs2 Rs 10 Rs
Ratios And ProportionsRatios And Proportions
1(10) = 2x1(10) = 2x 10 = 2x10 = 2x 1010 = = 2x2x 2 22 2 5 = x5 = x
You can buy 5 toffeesYou can buy 5 toffees
Ratios And ProportionsRatios And Proportions
Q)Q) Jasmine bought 32 kiwi fruit for Jasmine bought 32 kiwi fruit for Rs.16.How many kiwi can Jasmine Rs.16.How many kiwi can Jasmine buy if she only had Rs. 4?buy if she only had Rs. 4?
Solution:Solution:
32 Kiwi32 Kiwi = = xx16 Rs 4 Rs16 Rs 4 Rs32(4)=16x32(4)=16x128 = 16x128 = 16x
128/16 = x128/16 = x 8=x8=x
Ratios And ProportionsRatios And ProportionsQ)A special cereal mixture contains rice, wheat and corn in the ratio of 2:3:5. Q)A special cereal mixture contains rice, wheat and corn in the ratio of 2:3:5.
If a bag of the mixture contains 3 pounds of rice, how much corn does it If a bag of the mixture contains 3 pounds of rice, how much corn does it contain? contain?
Solutions:Solutions: Assign variables :Assign variables :
Let Let x x = amount of corn= amount of cornWrite the items in the ratio as a fraction.Write the items in the ratio as a fraction.
Cross MultiplicationCross Multiplication2 × 2 × xx = 3 × 5 = 3 × 5
22xx = 15 = 15
The mixture contains 7.5 pounds of corn. The mixture contains 7.5 pounds of corn.
InequalitiesInequalitiesInIn
One VariableOne Variable
Inequalities In one VariableInequalities In one Variable
Mathematical equations containing one Mathematical equations containing one or more of these or more of these (>, ≥ , < , ≤ )(>, ≥ , < , ≤ ) symbols are called inequality.symbols are called inequality.
Properties used to solve inequalities:Properties used to solve inequalities:1. If a>b then a+c>b+c.1. If a>b then a+c>b+c.2. If a>b then a-c>b-c.2. If a>b then a-c>b-c.3. If a>b and c>0, then ac>bc.3. If a>b and c>0, then ac>bc.4. If a>b and c>0, then a/c > b/c.4. If a>b and c>0, then a/c > b/c.
Inequalities In one VariableInequalities In one VariableQ) Solve and graph the solution set of:Q) Solve and graph the solution set of:
3(23(2x x + 4) > 4+ 4) > 4x x + 10+ 10 Solution:Solution:
66x x + 12 > 4+ 12 > 4x x + 10+ 10
Subtract 4Subtract 4xx from both sides. from both sides.22x x + 12 > 10+ 12 > 10
Subtract 12 from both sides.Subtract 12 from both sides.22xx > -2 > -2
Divide both sides by 2, but don't change the direction of Divide both sides by 2, but don't change the direction of the inequality, since we didn't divide by a negative.the inequality, since we didn't divide by a negative.
xx > -1 > -1 Open circle at -1 (since Open circle at -1 (since x x can not equal -1) and an arrow to can not equal -1) and an arrow to
the right (because we want values larger than -1).the right (because we want values larger than -1).
Inequalities In one VariableInequalities In one VariableQ) Solve and graph the solution set of:Q) Solve and graph the solution set of:
5 - 35 - 3x x ≤ 13 + ≤ 13 + xxSolution:Solution:
Subtract 5 from both sides.Subtract 5 from both sides.-3-3xx ≤ 8 + ≤ 8 + xx
Subtract Subtract xx from both sides. from both sides.-4-4xx ≤ 8 ≤ 8
divide both sides by -4,divide both sides by -4,and don't forget to change the direction of the and don't forget to change the direction of the
inequality ! inequality ! (We divided by a negative.)(We divided by a negative.)
xx ≥ -2 ≥ -2 Closed circle at -2 (since Closed circle at -2 (since x x can equal -2) and an can equal -2) and an arrow to the right (because we want values larger arrow to the right (because we want values larger
than -2).than -2).
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