Solving Equations
Solving equations is a recurring theme in much of mathematics.
We will study some of the principles used to solve equations and
then use equations to solve applied problems.
Solving equations is essential for problem solving in algebra.
In this section, we study two of the most important principles
used for this task.
Equations and Solutions
We have already seen that an equation is a number sentence
stating that the expressions on either side of the equals sign
represent the same number. Some equations, like 3 + 2 = 5 or 2x +
6 = 2(x + 3) are always true and some, like 3 + 2 = 6 or x + 2 =
x + 3 are never true. In this text, we will concentrate on
equations like 3x + 6 = 17 or 7x = 141 that are sometimes true,
depending on the replacement value for the variable.
Solution of an Equation: Any replacement for
the variable that makes an equation true is called a solution of
the equation. To solve an equation means to find all of its
solutions.
To determine whether a number is a solution, we substitute
that number for the variable throughout the equation. If the
values on both sides of the equals sign are the same, then the
number that was substituted is a solution.
Example 1
Determine whether 7 is a solution of x + 6 = 13
Solution
We have
Writing the equation.
Substituting 7 for x.
Note that 7, not 13, is the solution.
Since the lefthand and the righthand sides are the same, 7
is a solution.
Example 2
Determine whether 19 is a solution of 7x = 141.
Solution
We have
Writing the equation.
Substituting 19 for x.
The statement 133 = 141 is false.
Since the lefthand and the righthand sides differ, 19 is not
a solution.
The Addition Principle
Consider the equation x = 7.
We can easily see that the solution of this equation is 7.
Replacing x with 7, we get 7 = 7, which is true.
Now consider the equation x + 6 = 13.
In Example 1, we found that the solution of x + 6 = 13 is also
7. Although the solution of x = 7 may seem more obvious, the
equations x + 6 = 13 and x = 7 are equivalent.
Equivalent Equations: Equations with the same
solutions are called equivalent equations.
There are principles that enable us to begin with one equation
and end up with an equivalent equation, like x = 7, for which the
solution is obvious. One such principle concerns addition. The
equation a = b says that a and b stand for the same number.
Suppose this is true, and some number c is added to a. We get the
same result if we add c to b, because a and b are the same
number.
The Addition Principle: For any real numbers
a, b, and c, a = b is equivalent to a + c = b + c.
To visualize the addition principle, consider a balance
similar to one a jeweler might use. (See the figure.) When the
two sides of the balance hold quantities of equal weight, the
balance is level. If weight is added or removed, equally, on both
sides, the balance will remain level.
When using the addition principle, we often say that we
“add the same number to both sides of an equation”. We
can also “subtract the same number from both sides”,
since subtraction can be regarded as the addition of an opposite.
Example 3
Solve: x + 5 = 7.
Solution
We can add any number we like to both sides. Since is the
opposite, or additive inverse, of 5, we add to each side:
x + 5 = 7
x + 5  5 = 7  5 Using the addition principle: adding 5 to
both sides or subtracting 5 from both sides.
x + 0 = 12 Simplifying: x + 5  5 = x + 5 + (5) = x + 0.
x = 12 Using the identity property of 0.
It is obvious that the solution of x = 12 is the number 12.
To check the answer in the original equation, we substitute.
Check:
The solution of the original equation is 12.
In Example 3, note that because we added the opposite, or
additive inverse, of 5, the left side of the equation simplified
to x plus the additive identity, 0, or simply x. These steps
effectively replaced the 5 on the left with a 0. When solving x +
a = b for x, we simply add a to (or subtract a from) both sides.
Example 4
Solve: 6.5 = y  8.4
The variable is on the right side this time. We can isolate y
by adding 8.4 to each side:
6.5 = y  8.4 This can be regarded as 6.5 = y + ( 8.4)
6.5 + 8.4 = y  8.4 + 8.4 Using the addition principle:
Adding 8.4 to both sides “eliminates” 8.4 on the right
side.
1.9 = y Sice y  8.4 + 8.4 = y + (8.4) + 8.4 = y + 0 = y
Check:
The solution is 1.9.
Note that the equations a = b and b = a have the same meaning.
Thus, 6.5 = y  8.4 could have been rewritten as y8.4 = 6.5.
Example 5
Solve:
Solution
We have
Adding
to
both sides.
Multiplying by 1 to obtain a common denominator
The check is left to the student. The solution is.
The Multiplication Principle
A second principle for solving equations concerns multiplying.
Suppose a and b are equal. If a and b are multiplied by some
number c , then ac and bc will also be equal.
The Multiplication Principle: For any real
numbers a , b , and c , with c 0, a = b
is equivalent to a.c = b.c.
