660 Chapter 8 An Introduction to Algebra
Self Check 3
Solve:
a.
b.
Now TryProblems 39 and 43 20 n 29 5 b 38Since 5 is obviously the solution of the equivalent equation , the solution of
the original equation, , is also 5. To check this result, we substitute 5 for
in the original equation and simplify.
This is the original equation.
Substitute 5 for.
True
Since the statement 33 is true, 5 is the solution ofx 2 3.3 3
5 2 3 xx 2 3x 2 3 xx 5The Language of Algebra We solve equations by writing a series of steps
that result in an equivalent equation of the form
or
We say the variable is isolatedon one side of the equation.Isolatedmeans
alone or by itself.xa number a numberxEXAMPLE (^3) Solve: a. b.
StrategyWe will use a property of equality to isolate the variable on one side of
the equation.
WHYTo solve the original equation, we want to find a simpler equivalent
equation of the form or , whose solution is obvious.
Solution
a.To isolate on the right side, we use the addition property of equality. We can
undo the subtraction of 7 by adding 7 to both sides.
This is the equation to solve.
Add 7 to both sides.
Check: This is the original equation.
Substitute for.
True
Since the statement is true, the solution is.
Caution! We may solve an equation so that the variable is isolated on either
side of the equation. Note that is equivalent to.
b.To isolate , we use the addition property of equality. We can eliminate on
the left side by adding its opposite to both sides.
The equation to solve.
Add 27 to both sides.
Check: This is the original equation.
Substitute 24 for.
True
The solution of 27 y 3 is 24.
3 3
27 24 3 y 27 y 3The sum of a number and itsy (^24) opposite is zero: 27 27 0.
27 y 27 3 27
27 y 3
y 27
12 y y 12
19 19 12
19 19
19 12 7 12 y 19 y 7On the left side, add. On the right side, the sum of
12 y a number and its opposite is zero: 7 7 0. 19 7 y 7 7 19 y 7yya number a numbery 19 y 7 27 y 3