Literal Equations

Given an equation with many variables, to solve for a particular

variable means to isolate that variable to one side of the

equation.

Literal Equations

Given an equation with many variables, to solve for a particular

variable means to isolate that variable to one side of the

equation.

Literal Equations

Example A.

a. Solve for x if x + b = c

b. Solve for w if yw = 5.

Given an equation with many variables, to solve for a particular

variable means to isolate that variable to one side of the

equation. We try to accomplish this just as solving equations

for the x, by +, – the same term, or * , / by the same quantity

to both sides of the equations.

Example A.

a. Solve for x if x + b = c

Literal Equations

b. Solve for w if yw = 5.

Given an equation with many variables, to solve for a particular

variable means to isolate that variable to one side of the

equation. We try to accomplish this just as solving equations

for the x, by +, – the same term, or * , / by the same quantity

to both sides of the equations.

Example A.

a. Solve for x if x + b = c

Remove b from the LHS by subtracting from both sides

x + b = c

–b –b

x = c – b

Literal Equations

b. Solve for w if yw = 5.

Example A.

a. Solve for x if x + b = c

Remove b from the LHS by subtracting from both sides

x + b = c

–b –b

x = c – b

Literal Equations

b. Solve for w if yw = 5.

Remove y from the LHS by dividing both sides by y.

Given an equation with many variables, to solve for a particular

variable means to isolate that variable to one side of the

equation. We try to accomplish this just as solving equations

for the x, by +, – the same term, or * , / by the same quantity

to both sides of the equations.

Example A.

a. Solve for x if x + b = c

Remove b from the LHS by subtracting from both sides

x + b = c

–b –b

x = c – b

Literal Equations

b. Solve for w if yw = 5.

Remove y from the LHS by dividing both sides by y.

yw = 5

yw/y = 5/y

Given an equation with many variables, to solve for a particular

variable means to isolate that variable to one side of the

equation. We try to accomplish this just as solving equations

for the x, by +, – the same term, or * , / by the same quantity

to both sides of the equations.

Example A.

a. Solve for x if x + b = c

Remove b from the LHS by subtracting from both sides

x + b = c

–b –b

x = c – b

5y

Literal Equations

b. Solve for w if yw = 5.

Remove y from the LHS by dividing both sides by y.

yw = 5

yw/y = 5/y

w =

Given an equation with many variables, to solve for a particular

variable means to isolate that variable to one side of the

equation. We try to accomplish this just as solving equations

for the x, by +, – the same term, or * , / by the same quantity

to both sides of the equations.

Literal EquationsAdding or subtracting a term to both sides may be viewed as

moving the term across the " = " and change its sign.

To solve for a specific variable in a simple literal equation, do

the following steps.

Literal EquationsAdding or subtracting a term to both sides may be viewed as

moving the term across the " = " and change its sign.

To solve for a specific variable in a simple literal equation, do

the following steps.

1. If there are fractions in the equations, multiple by the

LCD to clear the fractions.

Literal EquationsAdding or subtracting a term to both sides may be viewed as

moving the term across the " = " and change its sign.

To solve for a specific variable in a simple literal equation, do

the following steps.

1. If there are fractions in the equations, multiple by the

LCD to clear the fractions.

2. Isolate the term containing the variable we wanted to

solve for

Literal EquationsAdding or subtracting a term to both sides may be viewed as

moving the term across the " = " and change its sign.

To solve for a specific variable in a simple literal equation, do

the following steps.

1. If there are fractions in the equations, multiple by the

LCD to clear the fractions.

2. Isolate the term containing the variable we wanted to

solve for – move all the other terms to other side of the

equation.

Literal EquationsAdding or subtracting a term to both sides may be viewed as

moving the term across the " = " and change its sign.

To solve for a specific variable in a simple literal equation, do

the following steps.

1. If there are fractions in the equations, multiple by the

LCD to clear the fractions.

2. Isolate the term containing the variable we wanted to

solve for – move all the other terms to other side of the

equation. 3. Isolate the specific variable by dividing the rest of the

factor to the other side.

Literal EquationsAdding or subtracting a term to both sides may be viewed as

moving the term across the " = " and change its sign.

To solve for a specific variable in a simple literal equation, do

the following steps.

1. If there are fractions in the equations, multiple by the

LCD to clear the fractions.

2. Isolate the term containing the variable we wanted to

solve for – move all the other terms to other side of the

equation. 3. Isolate the specific variable by dividing the rest of the

factor to the other side.

Literal Equations

Example B.

a. Solve for x if (a + b)x = c

Adding or subtracting a term to both sides may be viewed as

moving the term across the " = " and change its sign.

To solve for a specific variable in a simple literal equation, do

the following steps.

1. If there are fractions in the equations, multiple by the

LCD to clear the fractions.

2. Isolate the term containing the variable we wanted to

solve for – move all the other terms to other side of the

equation. 3. Isolate the specific variable by dividing the rest of the

factor to the other side.

Literal Equations

Example B.

a. Solve for x if (a + b)x = c

(a + b) x = c div the RHS by (a + b)

Adding or subtracting a term to both sides may be viewed as

moving the term across the " = " and change its sign.

To solve for a specific variable in a simple literal equation, do

the following steps.

1. If there are fractions in the equations, multiple by the

LCD to clear the fractions.

2. Isolate the term containing the variable we wanted to

solve for – move all the other terms to other side of the

equation. 3. Isolate the specific variable by dividing the rest of the

factor to the other side.

Literal Equations

Example B.

a. Solve for x if (a + b)x = c

(a + b) x = c

x =c

(a + b)

div the RHS by (a + b)

Adding or subtracting a term to both sides may be viewed as

moving the term across the " = " and change its sign.

b. Solve for w if 3y2w = t – 3

Literal Equations

b. Solve for w if 3y2w = t – 3

3y2w = t – 3 div the RHS by 3y2

Literal Equations

b. Solve for w if 3y2w = t – 3

3y2w = t – 3

w =t – 3 3y2

div the RHS by 3y2

Literal Equations

b. Solve for w if 3y2w = t – 3

3y2w = t – 3

c. Solve for a if b2 – 4ac = 5

w =t – 3 3y2

div the RHS by 3y2

Literal Equations

b. Solve for w if 3y2w = t – 3

3y2w = t – 3

c. Solve for a if b2 – 4ac = 5

b2 – 4ac = 5

w =t – 3 3y2

div the RHS by 3y2

move b2 to the RHS

Literal Equations

b. Solve for w if 3y2w = t – 3

3y2w = t – 3

c. Solve for a if b2 – 4ac = 5

b2 – 4ac = 5

– 4ca = 5 – b2

w =t – 3 3y2

div the RHS by 3y2

move b2 to the RHS

Literal Equations

b. Solve for w if 3y2w = t – 3

3y2w = t – 3

c. Solve for a if b2 – 4ac = 5

b2 – 4ac = 5

– 4ca = 5 – b2

w =t – 3 3y2

div the RHS by 3y2

move b2 to the RHS

div the RHS by –4c

Literal Equations

b. Solve for w if 3y2w = t – 3

3y2w = t – 3

c. Solve for a if b2 – 4ac = 5

b2 – 4ac = 5

– 4ca = 5 – b2

w =t – 3 3y2

a = 5 – b2

–4c

div the RHS by 3y2

move b2 to the RHS

div the RHS by –4c

Literal Equations

b. Solve for w if 3y2w = t – 3

3y2w = t – 3

c. Solve for a if b2 – 4ac = 5

b2 – 4ac = 5

– 4ca = 5 – b2

w =t – 3 3y2

a = 5 – b2

–4c

div the RHS by 3y2

move b2 to the RHS

div the RHS by –4c

Literal Equations

d. Solve for y if a(x – y) = 10

b. Solve for w if 3y2w = t – 3

3y2w = t – 3

c. Solve for a if b2 – 4ac = 5

b2 – 4ac = 5

– 4ca = 5 – b2

w =t – 3 3y2

a = 5 – b2

–4c

div the RHS by 3y2

move b2 to the RHS

div the RHS by –4c

Literal Equations

d. Solve for y if a(x – y) = 10

a(x – y) = 10 expand

b. Solve for w if 3y2w = t – 3

3y2w = t – 3

c. Solve for a if b2 – 4ac = 5

b2 – 4ac = 5

– 4ca = 5 – b2

w =t – 3 3y2

a = 5 – b2

–4c

div the RHS by 3y2

move b2 to the RHS

div the RHS by –4c

Literal Equations

d. Solve for y if a(x – y) = 10

a(x – y) = 10 expand

ax – ay = 10

b. Solve for w if 3y2w = t – 3

3y2w = t – 3

c. Solve for a if b2 – 4ac = 5

b2 – 4ac = 5

– 4ca = 5 – b2

w =t – 3 3y2

a = 5 – b2

–4c

div the RHS by 3y2

move b2 to the RHS

div the RHS by –4c

Literal Equations

d. Solve for y if a(x – y) = 10

a(x – y) = 10 expand

ax – ay = 10 subtract ax

b. Solve for w if 3y2w = t – 3

3y2w = t – 3

c. Solve for a if b2 – 4ac = 5

b2 – 4ac = 5

– 4ca = 5 – b2

w =t – 3 3y2

a = 5 – b2

–4c

div the RHS by 3y2

move b2 to the RHS

div the RHS by –4c

Literal Equations

d. Solve for y if a(x – y) = 10

a(x – y) = 10 expand

ax – ay = 10 subtract ax

– ay = 10 – ax

b. Solve for w if 3y2w = t – 3

3y2w = t – 3

c. Solve for a if b2 – 4ac = 5

b2 – 4ac = 5

– 4ca = 5 – b2

w =t – 3 3y2

a = 5 – b2

–4c

div the RHS by 3y2

move b2 to the RHS

div the RHS by –4c

Literal Equations

d. Solve for y if a(x – y) = 10

a(x – y) = 10 expand

ax – ay = 10 subtract ax

– ay = 10 – ax div by –a

b. Solve for w if 3y2w = t – 3

3y2w = t – 3

c. Solve for a if b2 – 4ac = 5

b2 – 4ac = 5

– 4ca = 5 – b2

w =t – 3 3y2

a = 5 – b2

–4c

div the RHS by 3y2

move b2 to the RHS

div the RHS by –4c

Literal Equations

d. Solve for y if a(x – y) = 10

a(x – y) = 10 expand

ax – ay = 10 subtract ax

– ay = 10 – ax div by –a

y = 10 – ax

–a

Multiply by the LCD to get rid of the denominator then solve.

Literal Equations

–4 =d

3d + b

Multiply by the LCD to get rid of the denominator then solve.

Example C.

Solve for d if

Literal Equations

–4 =d

3d + b

Multiply by the LCD to get rid of the denominator then solve.

multiply by the LCD d– 4 =d

3d + b

Example C.

Solve for d if

Literal Equations

–4 =d

3d + b

Multiply by the LCD to get rid of the denominator then solve.

multiply by the LCD d

d

– 4 =d

3d + b

– 4 = d3d + b

( )d

Example C.

Solve for d if

Literal Equations

–4 =d

3d + b

Multiply by the LCD to get rid of the denominator then solve.

multiply by the LCD d

d

– 4 =d

3d + b

– 4 =d

3d + b( )

d

Example C.

Solve for d if

Literal Equations

–4 =d

3d + b

Multiply by the LCD to get rid of the denominator then solve.

multiply by the LCD d

d

– 4 =d

3d + b

– 4 =d

3d + b( )

d

– 4d = 3d + b

Example C.

Solve for d if

Literal Equations

–4 =d

3d + b

Multiply by the LCD to get rid of the denominator then solve.

multiply by the LCD d

d

– 4 =d

3d + b

– 4 =d

3d + b( )

d

– 4d = 3d + b move –4d and b

Example C.

Solve for d if

Literal Equations

–4 =d

3d + b

Multiply by the LCD to get rid of the denominator then solve.

multiply by the LCD d

d

– 4 =d

3d + b

– 4 =d

3d + b( )

d

– 4d = 3d + b

– b = 3d + 4d

move –4d and b

Example C.

Solve for d if

Literal Equations

–4 =d

3d + b

Multiply by the LCD to get rid of the denominator then solve.

multiply by the LCD d

d

– 4 =d

3d + b

– 4 =d

3d + b( )

d

– 4d = 3d + b

– b = 3d + 4d

move –4d and b

– b = 7d

Example C.

Solve for d if

Literal Equations

–4 =d

3d + b

Multiply by the LCD to get rid of the denominator then solve.

multiply by the LCD d

d

– 4 =d

3d + b

– 4 =d

3d + b( )

d

– 4d = 3d + b

– b = 3d + 4d

move –4d and b

– b = 7d

= d7

–b

Example C.

Solve for d if

Literal Equations

div. by 7

Flipping Equations

If it’s advantageous to do so,

we may reposition the equation

by reciprocating both sides of

an equation.

Literal Equations

x = 3 +12y

Example D.

Solve for y if

Flipping Equations

Literal Equations

Flipping Equations

Literal Equations

x = 3 +12y

Example D.

Solve for y if

Flipping Equations

If it’s advantageous to do so,

we may reposition the equation

by reciprocating both sides of

an equation.

Literal Equations

x = 3 +12y

Example D.

Solve for y if

=DC

Flipping Equations

If it’s advantageous to do so,

we may reposition the equation

by reciprocating both sides of

an equation. In particular:

Literal Equations

BA

=CD

AB

Reciprocate

both sides

x = 3 +12y

Example D.

Solve for y if

=DC

Flipping Equations

If it’s advantageous to do so,

we may reposition the equation

by reciprocating both sides of

an equation. In particular:

x = 3 +12y

Example D.

Solve for y if

Literal Equations

BA

=CD

AB

x = 3 +12y

Reciprocate

both sides

=DC

Flipping Equations

If it’s advantageous to do so,

we may reposition the equation

by reciprocating both sides of

an equation. In particular:

x = 3 +12y

Example D.

Solve for y if

Literal Equations

BA

=CD

AB

x = 3 +12y

x – 3 =12y

Reciprocate

both sides

=DC

Flipping Equations

If it’s advantageous to do so,

we may reposition the equation

by reciprocating both sides of

an equation. In particular:

x = 3 +12y

Example D.

Solve for y if

Literal Equations

BA

=CD

AB

x = 3 +12y

x – 3 =12y

reciprocating both sides,

=12y

x – 31

Reciprocate

both sides

=DC

Flipping Equations

If it’s advantageous to do so,

we may reposition the equation

by reciprocating both sides of

an equation. In particular:

x = 3 +12y

Example D.

Solve for y if

Literal Equations

BA

=CD

AB

x = 3 +12y

x – 3 =12y

reciprocating both sides, then div. by 2

=12y

x – 31

=2y

2(x – 3)1

Reciprocate

both sides

2

=DC

Flipping Equations

If it’s advantageous to do so,

we may reposition the equation

by reciprocating both sides of

an equation. In particular:

x = 3 +12y

Example D.

Solve for y if

Literal Equations

BA

=CD

AB

x = 3 +12y

x – 3 =12y

reciprocating both sides, then div. by 2

=12y

x – 31

=2y

2(x – 3)1

Reciprocate

both sides

2= y or

2(x – 3)1

y =

Exercise. Solve for the indicated variables.

Literal Equations

1. a – b = d – e for b. 2. a – b = d – e for e.

3. 2*b + d = e for b. 4. a*b + d = e for b.

5. (2 + a)*b + d = e for b. 6. 2L + 2W = P for W

7. (3x + 6)y = 5 for y 8. 3x + 6y = 5 for y

w =t – 3

613. for t w =

t – b a

14. for t

w =11. for t w =12. for tt6

6t

w =3t – b

a15. for t 16. (3x + 6)y = 5 for x

w =t – 3

617. + a for t w =

at – b 5

18. for t

w =at – b

c19. + d for t 3 =

4t – b t

20. for t

9. 3x + 6xy = 5 for y 10. 3x – (x + 6)y = 5z for y