# 2 8 study guide and intervention literal equations and dimensional analysis solve variables

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.