Name: ________________
Algebra 1
WORKSHEET: Literal Equations & Formulas
Oct. 15, 2009
Literal Equations
Simple linear equations such as 5𝑥 − 3 = 7 and −3𝑥 + 1 = 2 have the general form 𝑎𝑥 +
𝑏 = 𝑐. The equation 𝑎𝑥 + 𝑏 = 𝑐 is called a literal equation because the numbers (the
coefficient and constant terms) have been replaced by letters. When you solve a literal
equation, you can use the result to solve any equation that has the same form as the
literal equation. Here’s how we can solve the literal equation 𝑎𝑥 + 𝑏 = 𝑐:
Solve 𝑎𝑥 + 𝑏 = 𝑐
for 𝑥.
𝑎𝑥 = 𝑐 − 𝑏
𝑥=
subtract 𝑏 from each side.
𝑐−𝑏
𝑎
divide both sides by 𝑎.
For this literal equation solution to be valid, we must assume that 𝑎 ≠ 0. (Why?)
We can use this solution to solve other equations of the same form, such as 3𝑥 − 1 = 8.
Solve 3𝑥 − 1 = 8
𝑥=
for 𝑥.
8−(−1)
3
substitute 3 for a, -1 for b, and 8 for c.
9
3
simplify.
𝑥= =3
Try these in class: solve the literal equation for x, then use the solution to find an
answer for each of the specific equations given to the right.
1. 𝑎 − 𝑏𝑥 = 𝑐
7 − 4𝑥 = −1
and
56 − 11𝑥 = 12
2. 𝑎𝑥 = 𝑏𝑥 + 𝑐
3𝑥 = 2𝑥 + 7
and
−6𝑥 = 2𝑥 − 12
4
𝑥
3
and
−7
𝑥
4
3.
𝑎
𝑥
𝑏
𝑐
+𝑏 =𝑑
+
11
3
=5
3
+4=2
Formulas
Formulas are encountered frequently, not only in mathematics, but in all the sciences,
engineering, economics & finance; almost any technical endeavor is likely to require the
use of formulas. Stated simply, a formula is an equation relating a set of two or more
variables. Very often the variables represent physical, measurable quantities and the
formula relates to some physical law or property.
For instance, in Geometry there is a formula that relates the circumference and diameter
of a circle; it says that the ratio (more on ratios next week…) of the circumference (𝑐) to
the diameter (𝑑) is always equal to the constant known as pi (𝜋) – about 3.14. Here’s
c

d
how we write this using algebra:
Since this formula is an equation we can multiply both sides by the variable 𝑑:
c
d  d  c  d
d
This gives another version of the formula, which says that circumference is equal to pi
times diameter. In this form, the formula is more useful – it allows us to calculate the
circumference if we know the diameter, which is usually easier to measure. For
example, if we know that the diameter of a circle is 10 cm, then we can use this formula
to find the circumference: c    d  (3.14)(10cm)  31.4cm . (Note that we have used
3.14 as an approximation for  .)
Since formulas are equations, all the properties of equations apply, and we can use
these properties (Multiplication and Addition Properties of Equality) to manipulate a
formula to “solve” for any of the variables. Here’s another example. The average (A) of
two numbers (𝑥 & 𝑦) is given by the formula
A
1
x  y 
2
Let’s say we wanted to rewrite this formula to express one of the numbers 𝑥 in terms of
the average 𝐴 and the other variable 𝑦 – we need to solve for the variable 𝑥:
1
x  y   x  y
2
2A  y  x  y  y  x
2 A  2
First we multiply both sides by two (to get rid of the ½); then we subtract 𝑦 from both
sides to get the 𝑥 by itself on one side of the = sign. The strategy here is to peel away
the factors and terms surrounding the 𝑥.
Solve these formulas for 𝑥. What strategy would you use?
1.
3m  x  k
2.
rx  t  s  t
3.
L  pxw  y
Name: ________________
Solve each of these formulas for 𝑥.
𝑥
4. 𝑠 = 5𝑥
5. 𝑝 = 4
3
𝑥
7. 𝑎 = 2𝑥 − 𝑐
8. 𝑏 = 𝑘 − 11𝑥
9. 𝑚𝑥 + 𝑏 = 𝑦
6. 𝑛 =
Solve the following familiar formulas for the indicated variable:
1
2
10. Area of a triangle: 𝐴 = 𝑏ℎ
solve for ℎ.
11. Area of a circle: 𝐴 = 𝜋𝑟 2
solve for 𝑟 2 .
12. Perimeter of a rectangle: 𝐴 = 2𝑙 + 2𝑤
13. Einstein’s formula: 𝐸 = 𝑚𝑐 2
solve for 𝑚.
solve for 𝑤.
14. The formula for the horsepower (H) of an N-cylinder engine is H 
D2 N
where D
2.5
is the displacement of the engine (a measure of its size). Solve this formula for 𝑁.
15. The formula R  0.00625t  3.85 can supposedly be used to estimate the worldrecord time (in minutes) for the 1500 meter run t years after the year 1930.
a. What would be the predicted record time for 2010? (Bonus points for anyone
who can find the actual present 1500 meter world record.)
b. Solve this formula for t.
c. If we believe this formula, in what year should the record fall to 2 minutes?
14. The Universal Gas Law, used in Physics and Chemistry, says that the product of the
pressure (P) and volume (V) of a gas is equal to its temperature (T) times a constant (k).
a. Write this formula algebraically. (What does “product” mean?)
b. Solve for pressure (P).
c. Solve for the constant (k).
Solve these random formulas for the indicated variable:
15. 𝐽 = 𝑏𝑧 + 5
𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑏.
1
17. 𝑊 = 4 (𝑑 − 𝑓)
𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑑.
19. 4𝑒 − 2𝑠 = 𝑏 + 𝑠
21. ℎ = 𝑎 + 𝑏(𝑛 − 1)
23. 𝑎𝑥 + 𝑥 = 𝑐𝑦 − 𝑑
𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑒.
𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑎.
𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥.
16. 𝑇 = 1 − 𝑚𝑘
𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑚.
18. 𝑅 = 7(ℎ + 𝑛) 𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑛.
20. 4𝑒 − 2𝑠 = 𝑏 + 𝑠
22. ℎ = 𝑎 + 𝑏(𝑛 − 1)
24. 𝑎𝑥 + 𝑏𝑥 = 𝑐𝑦 − 𝑑𝑦
𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑠.
𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑛.
𝑠𝑜𝑙𝑣𝑒 𝑓𝑜𝑟 𝑥.