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 Dependent Variable

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# 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.