2-8 Study Guide and Intervention
Literal Equations and Dimensional Analysis

Solve for Variables Sometimes you may want to solve an equation such as V = ℓwh for one of its variables. For example, if you know the values of V, w, and h, then the equation ℓ =  is more useful for finding the value of ℓ. If an equation that contains more than one variable is to be solved for a specific variable, use the properties of equality to isolate the specified variable on one side of the equation.

Example 1: Solve 2x – 4y = 8, for y.
    2x – 4y = 8
2x – 4y – 2x = 8 – 2x
    –4y = 8 – 2x
       =
    y =   or 
The value of y i s .

Example 2: Solve 3m – n = km – 8, for m.
      3m – n = km – 8
     3m – n – km = km – 8 – km
     3m – n – km =  –8
3m – n – km + n =  –8 + n
    3m – km =  –8 + n
    m(3 – k) =  –8 + n
        =
     m =  or 
The value of m is . Since division by 0 is undefined, 3 – k ≠ 0, or k ≠ 3.
Exercises
Solve each equation or formula for the variable indicated.

    1. ax – b = c, for x            2. 15x + 1 = y, for x            3. (x + f) + 2 = j, for x

    4. xy + w = 9, for y            5. x(4 – k) = p, for k            6. 7x + 3y = m, for y

    7. 4(r + 3) = t, for r            8. 2x + b = w, for x            9. x(1 + y) = z, for x

10. 16w + 4x = y, for x            11. d = rt, for r                12. A =  , for h

13. C =  (F – 32), for F            14. P = 2ℓ + 2w, for w            15. A = ℓw, for ℓ
 

2-8 Study Guide and Intervention (continued)
Literal Equations and Dimensional Analysis

Use Formulas Many real-world problems require the use of formulas. Sometimes solving a formula for a specified variable will help solve the problem.

Example: The formula C = πd represents the circumference of a circle, or the distance around the circle, where
d is the diameter. If an airplane could fly around Earth at the equator without stopping, it would have traveled about 24,900 miles. Find the diameter of Earth.
C = πd                Given formula
d =                 Solve for d.
d =             Use π = 3.14.
d ≈ 7930            Simplify.
The diameter of Earth is about 7930 miles.

Exercises
    1. GEOMETRY The volume of a cylinder V is given by the formula V = h, where r is the radius and h is the height.

    a. Solve the formula for h.

    b. Find the height of a cylinder with volume 2500π cubic feet and radius 10 feet.

    2. WATER PRESSURE The water pressure on a submerged object is given by P = 64d, where P is the pressure in pounds per square foot, and d is the depth of the object in feet.

    a. Solve the formula for d.

    b. Find the depth of a submerged object if the pressure is 672 pounds per square foot.

    3. GRAPHS The equation of a line containing the points (a, 0) and (0, b) is given by the formula   +   = 1.

    a. Solve the equation for y.

    b. Suppose the line contains the points (4, 0), and (0, –2). If x = 3, find y.

    4. GEOMETRY The surface area of a rectangular solid is given by the formula x = 2ℓw + 2ℓh + 2wh,
where ℓ = length, w = width, and h = height.

    a. Solve the formula for h.

    b. The surface area of a rectangular solid with length 6 centimeters and width 3 centimeters is 72 square centimeters. Find the height.