Literal Equations
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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 – 3Literal 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 = d3d + b
Multiply by the LCD to get rid of the denominator then solve.
Example C. Solve for d if
Literal Equations
–4 = d3d + b
Multiply by the LCD to get rid of the denominator then solve.
multiply by the LCD d– 4 = d3d + b
Example C. Solve for d if
Literal Equations
–4 = d3d + b
Multiply by the LCD to get rid of the denominator then solve.
multiply by the LCD d
d
– 4 = d3d + b
– 4 = d3d + b( )
d
Example C. Solve for d if
Literal Equations
–4 = d3d + b
Multiply by the LCD to get rid of the denominator then solve.
multiply by the LCD d
d
– 4 = d3d + b
– 4 = d3d + b( )
d
Example C. Solve for d if
Literal Equations
–4 = d3d + b
Multiply by the LCD to get rid of the denominator then solve.
multiply by the LCD d
d
– 4 = d3d + b
– 4 = d3d + b( )
d
– 4d = 3d + b
Example C. Solve for d if
Literal Equations
–4 = d3d + b
Multiply by the LCD to get rid of the denominator then solve.
multiply by the LCD d
d
– 4 = d3d + b
– 4 = d3d + b( )
d
– 4d = 3d + b move –4d and b
Example C. Solve for d if
Literal Equations
–4 = d3d + b
Multiply by the LCD to get rid of the denominator then solve.
multiply by the LCD d
d
– 4 = d3d + b
– 4 = d3d + b( )
d
– 4d = 3d + b
– b = 3d + 4d
move –4d and b
Example C. Solve for d if
Literal Equations
–4 = d3d + b
Multiply by the LCD to get rid of the denominator then solve.
multiply by the LCD d
d
– 4 = d3d + b
– 4 = d3d + b( )
d
– 4d = 3d + b
– b = 3d + 4d
move –4d and b
– b = 7d
Example C. Solve for d if
Literal Equations
–4 = d3d + b
Multiply by the LCD to get rid of the denominator then solve.
multiply by the LCD d
d
– 4 = d3d + b
– 4 = d3d + b( )
d
– 4d = 3d + b
– b = 3d + 4d
move –4d and b
– b = 7d
= d 7–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 EquationsLiteral Equations
Flipping EquationsLiteral 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
Reciprocateboth 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
Reciprocateboth 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
Reciprocateboth 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
Reciprocateboth 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
= 2y2(x – 3)
1
Reciprocateboth 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
= 2y2(x – 3)
1
Reciprocateboth sides
2 = y or 2(x – 3)1y =
= 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
= 2y2(x – 3)
1
Reciprocateboth sides
2 = y or 2(x – 3)1y =
Be careful with the flip,
1y
+1x
11y
+1x
reciprocates
(which is not x + 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
a14. 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
518. for t
w = at – b c19. + d for t 3 = 4t – b
t20. for t
9. 3x + 6xy = 5 for y 10. 3x – (x + 6)y = 5z for y
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