Linear Equations in One Variable
An equation is a sentence that expresses the equality of two algebraic
expressions. Consider the equation
2x + 1 = 7.
Because 2(3) + 1 = 7 is true, we say that 3 satisfies the equation. No
other number in place of x will make the statement 2x + 1 = 7 true. However, an
equation might be satisfied by more than one number. For example, both 3 and -3
satisfy x2 = 9. Any number that satisfies an equation is called a
solution or root to the equation.
Solution Set
The set of all solutions to an equation is called the solution set to
the equation.
The solution set to 2x + 1 = 7 is {3}. To determine whether a number is in
the solution set to an equation, we simply replace the variable by the number
and see whether the equation is correct.
Example
Satisfying an equation
Determine whether each equation is satisfied by the number following the
equation.
a) 3x + 7 = -8, -5
b) 2(x - 1) = 2x + 3, 4
Solution
a) Replace x by -5 and evaluate each side of the equation.
| 3x + 7 |
= -8 |
|
| 3(-5) + 7 |
= -8 |
|
| -15 + 7 |
= -8 |
|
| -8 |
= -8 |
Correct |
Because -5 satisfies the equation, -5 is in the solution set to the equation.
b) Replace x by 4 and evaluate each side of the equation
| 2(x - 1) |
= 2x + 3 |
|
| 2(4 - 1) |
= 2(4) + 3 |
Replace x by 4. |
| 2(3) |
= 8 + 3 |
|
| 6 |
= 11 |
Incorrect |
The two sides of the equation have different values when x = 4. So 4 is not
in the solution set to the equation.
