3.8 Solving for a Variable

  STEPS Example Problem

Step #1

  Solve 3x – 4y = 7 for y

Step #2

   

Step #3

   

Step #4

   

Identify which variable you are solving for

Find all terms that have that variable

-4y

Move all terms with the variable to one side of the equation by adding or subtracting

Already done.

3x-4y=7

Move all terms without the variable to the opposite side of the equation by adding or subtracting

3x – 4y = 7

-3x -3x

– 4y = 7 - 3x

Step #5

   

Step #6

   

Step #7

   

Use the distributive property to factor terms with the variable in question

 (not always required)

Not Required

Undo what the variable is multiplied or divided by 4

37

x

y

Simplify (if possible)

4

73

xy

Practice

5

129 c

5b + 12c = 9

Solve for b. 

Solve for c. 

y = mx + b

Solve for m. 

Solve for b. 

5b=9-12c

b=

c=9-5b

c=12

59 b

y-b=mx

(y-b)/x=my-mx=b

Factoring: undo distributive property (take out what all terms have in common)

1) xm + ym = 2) 7t + xt – rt =

3) 5y – 5t + 5yt= 4) 3y + rty – 5xy =

m(x+y) t(7+x-r)

5(y-t+y)y(3+rt-5x)

Practice

s

t

2

5

4

g

Solve 2m - t = 3m + 5 for m. 

Solve 5g + h = g for g. 

Solve 2m - t = sm + 5 for m. 

Solve at + b = ar – c for a.  

-t=1m+5

-t-5=m

h=-4g

h=

rt

bc

2m-t-sm=52m-sm=5+tm(2-s)=5+t

m=

at+b-ar=-cat-ar=-c-b

a(t-r)=-c-b

a=

Solving with fractions The only additional step from the original

steps for solving an equation for a given variable is to _________________ as your initial step.

Cross multiply

Solving with fractions

cd

5

43

3

5c 4 d

2

b-3a

3

1-bSolve for c.

  

Solve for b.    

3d=4+5c

3d-4=5c

5

29

ab

2(b-1)=3(3a-b)

2b-2=9a-3b

5b-2=9a

5b=9a+2

On your own:

3

2a b- 8

2

r-5

s

2r

22

3

nbxy

s

6

yd

Solve for a.  

Solve for r.   

Solve for b.  

Solve for y.   

Translate and solveThree less than a number t equals another number s plus 7. Solve for s.  

The sum of a number x and 2 divided by 3 is as much as another number y minus 4. Solve for x.

The area of a triangle is equal to one-half the product of base b and height h. Solve for height.   

The formula for the circumference of a circle, C, is the product 2π and the radius of the circle r. Solve for r.  

T-3=s+7

T-10=s

Example 1: The formula s = ½ at2 represents the distance s that a free-falling

object will fall near a planet or the moon in a given time t. In the formula, a represents the acceleration due to gravity.

a)  Solve the formula for a.

b)  A free – falling object near the moon drops 20.5 meters in 5 seconds. What is the value of acceleration for the moon?

Example 2: The formula for the volume of a cylinder is V = πr2h,

where r is the radius of the cylinder and h is the height.

a)  Solve the formula for h.

b)  What is the height of a cylindrical swimming pool that has a radius of 12 feet and a volume of 1810 cubic feet?

Example 3: In uniform circular motion, the speed v of a point on the edge of a

spinning disk is , where r is the radius of the disk and T is the time it takes the point to travel around the circle once.

a)  Solve the formula for r.

b)  Suppose a merry-go-round is spinning once every 3 seconds. If a point on the outside edge has a speed of 12.56 feet per second, what is the radius of the merry-go-round? (Use 3.14 for π)

rT

2 v

Example 4: A car’s fuel economy E (miles per gallon) is given by the formula

, where m is the number of miles driven and g is the number of gallons of fuel.

a)  Solve the formula for miles, m.

b)  If Mr. Smith’s car has an average gas consumption of 30 miles per gallon and he used 9.5 gallons of gas, how far did he drive?

c) If Josh’s car gets 25 miles per gallon and he drove for 215 miles, how much gas did his car use?

gE

m

Eg=m

E=9.5, m=30 9.5g=30