Solving linear equations (chapter 2)

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 .


For example there is a term:For example there is a term:

+2 - 5-6 3x x

−4x −2


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


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.


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


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)


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.


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


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


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


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.


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


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


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


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


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).

Solving linear equations (chapter 2)

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