Chapter Four — Linear Equations in One Variable
The Humongous Book of Algebra Problems
73
Equations Containing Multiple Variables
Equations with TWO variables (like x and y) or more
4.37 Solve , the formula for the lateral surface a right circular cylinder,
for r.
Whether an equation contains one variable or many, the same technique is
employed to solve for (or isolate) a variable. To isolate r on the left side of the
equation, eliminate the values by which r is multiplied.
4.38 Solve , the formula that converts degrees Celsius to degrees
Fahrenheit, for C.
Isolate C left of the equal sign by subtracting 32 from both sides of the equation
and then multiplying every term by .
The equivalent equation , obtained by applying the distributive
property, represents another acceptable answer.
Note: Problems 4.39–4.40 refer to the equation 5x – 3y = 30.
4.39 Solve the equation for x.
To isolate x, add 3y to both sides of the equation and then divide each term by
5, the coefficient of x.
So, r is
multiplied by
three things—
two numbers (2
and
) and a
variable (h). Either
divide both sides by
or multiply both
sides by
to
move them
away from r.
The reciprocal
of C’s coefcient.
Move the
non x–term to the
other side of the
equation before
you deal with x’s
coefcient.
It’s a good idea to write variable terms
before number terms. You can pull that y out in front
of its fraction because
.
Chapter Four — Linear Equations in One Variable
The Humongous Book of Algebra Problems
74
Note: Problems 4.39–4.40 refer to the equation 5x – 3y = 30.
4.40 Solve the equation for y.
To isolate y, subtract 5x from both sides of the equation and then divide each
term by –3, the coefficient of y.
Note: Problems 4.41–4.42 refer to the equation a – (b + 4) = 2a + 3b – 6.
4.41 Solve the equation for a.
Apply the distributive property to the left side of the equation.
a – b – 4 = 2a + 3b – 6
Move all of the a–terms, the terms that contain the variable for which you are
solving, left of the equal sign by subtracting 2a from both sides of the equation
and combine like terms. In this case, a – 2a = 1a – 2a = –1a = –a.
Move all terms not containing a to the right side of the equation and simplify
the result.
Multiply each term of the equation by –1 to isolate a left of the equal sign and
thereby solve the equation for a.
Think of
–(b + 4) as
(–1)(b + 4):
Chapter Four — Linear Equations in One Variable
The Humongous Book of Algebra Problems
75
Note: Problems 4.41–4.42 refer to the equation a – (b + 4) = 2a + 3b – 6.
4.42 Solve the equation for b.
Use the technique described in Problem 4.41, this time isolating b left of the
equal sign.
4.43 Solve the equation xy + 2 = –7y(x + 1) for x. (Assume y ≠ 0.)
Apply the distributive property.
Move all terms containing x, the variable for which you are solving, to the left
side of the equation. Move all other terms right of the equal sign.
To isolate (and, therefore, solve the equation for) x, divide each term in the
equation by 8y.
xy and 7xy
are like terms
because their
variables match—
they both contain xy.
Combining like terms
means combining
their coefcients:
xy + 7xy = 1xy +
7xy = 8xy.
You can’t write
as .
You can only pull y out
of the fraction when
it’s in the numerator,
and this y is in the
denominator.
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