Exercise 106 Page 323

Practice makes perfect

To solve an equation, we should first gather all of the variable terms on one side and all of the constant terms on the other side using the properties of equality. In this case, we can start by using the Multiplication Property of Equality to get rid of the fractions.

1/4x+1/10=- 3/20x-7/10

MultEqn

LHS * 20=RHS* 20

20(1/4x+1/10)=20(- 3/20x-7/10)

Distr

Distribute 20

20(1/4x)+20(1/10)=20(- 3/20x)-20(7/10)

MoveLeftFacToNum

a*b/c= a* b/c

20(1)/4x+20(1)/10=- 20(3)/20x-20(7)/10

Multiply

20/4x+20/10=- 60/20x-140/10

CalcQuot

Calculate quotient

5x+2=- 3x-14

Now we can use the properties of equality to group the variable and constant terms together.

5x+2=- 3x-14

AddEqn

LHS+3x=RHS+3x

8x+2=- 14

SubEqn

LHS-2=RHS-2

8x=- 16

DivEqn

.LHS /8.=.RHS /8.

8x/8=- 16/8

CalcQuot

Calculate quotient

x=- 2

To solve this equation, we will first gather all of the variable terms on the left-hand side and all of constant terms on the right-hand side of the equation using the properties of equality.

3x+4.5=4.5x-18

SubEqn

LHS-4.5x=RHS-4.5x

-1.5x+4.5=- 18

SubEqn

LHS-4.5=RHS-4.5

- 1.5x=- 22.5

Now we will divide the equation by - 1.5 to isolate x.

- 1.5x=- 22.5

DivEqn

.LHS /(- 1.5).=.RHS /(- 1.5).

- 1.5x/-1.5=- 22.5/- 1.5

CalcQuot

Calculate quotient

x=15

To solve the given equation, we should first gather all of the variable terms on one side and all of the constant terms on the other side using the properties of equality. Here, we will first multiply the equation by 12 to get rid of the fractions.

1/12x-1/12= 1/8x+1/6-1/6x

MultEqn

LHS * 24=RHS* 24

24(1/12x-1/12)= 24(1/8x+1/6-1/6x)

Distr

Distribute 24

24(1/12x)-24(1/12)= 24(1/8x)+24(1/6)-24(1/6x)

MoveLeftFacToNum

a*b/c= a* b/c

24(1)/12x-24(1)/12= 24(1)/8x+24(1)/6-24(1)/6x

IdPropMult

Identity Property of Multiplication

24/12x-24/12= 24/8x+24/6-24/6x

CalcQuot

Calculate quotient

2x-2=3x+4-4x

SubTerm

Subtract term

2x-2=4-x

Next, we can use the properties of equality to gather all the x-terms on the left-hand side and the constant terms on the right-hand side of the equation. From there, we will continue to simplify until we have isolated x.

2x-2=4-x

AddEqn

LHS+x=RHS+x

3x-2=4

AddEqn

LHS+2=RHS+2

3x=6

DivEqn

.LHS /3.=.RHS /3.

3x/3=6/3

CalcQuot

Calculate quotient

x=2

To solve the given equation, we will gather all of the x-terms on the left-hand side and all of the constant terms on the right-hand side of the equation using the properties of equality.

6.25x+7.5-2.5x=3.75x-8.75

SubTerm

Subtract term

3.75x+7.5=3.75x-8.75

SubEqn

LHS-3.75x=RHS-3.75x

7.5≠ - 8.75 *

We got a contradiction, as 7.5 can never be equal to -8.75. This means that the given equation has no solution.