Problem

Solve the equation, and check the solution. See ExamplesEXAMPLE Solving a Linear EquationS...

Solve the equation, and check the solution. See Examples

EXAMPLE Solving a Linear Equation
Solve −6x + 5 = 17.
Step 1 There are no parentheses, fractions, or decimals in this equation, so this step is not necessary.
Step 4 Check by substituting −2 for x in the original equation.
The check confirms that −2 is the solution. The solution set is {−2}.
EXAMPLE Solving a Linear Equation
Solve 3x + 2 = 5x − 8.
Step 1 There are no parentheses, fractions, or decimals in the equation.
Step 4 Check by substituting 5 for x in the original equation.
The check confirms that 5 is the solution. The solution set is {5}.
EXAMPLE Solving a Linear Equation
Solve 4 (k − 3) − k = k − 6.
Step 1 Clear parentheses using the distributive property.
The solution set is {3}.
EXAMPLE Solving a Linear Equation
Solve 8z − (3 + 2z) = 3z + 1.
Step 4 Check that is the solution set.
EXAMPLE Solving a Linear Equation
Solve 4(4 − 3x) = 32 − 8(x + 2).
Because a true statement results, the solution set is {0}.
EXAMPLE Solving an Equation That Has Infinitely Many Solutions
Solve 5x − 15 = 5(x − 3).
Solution set: {all real numbers}
Because the last statement (0 = 0) is true, any real number is a solution. We could have predicted this from the second line in the solution.
Try several values for x in the original equation to see that they all satisfy it.
An equation with both sides exactly the same, like 0 = 0, is an identity. An identity is true for all replacements of the variables. As shown above, we write the solution set as
EXAMPLE Solving an Equation That Has No Solution
Solve 2x + 3(x + 1) = 5x + 4.
A false statement (3 = 4) results. A contradiction is an equation that has no solution. Its solution set is the empty set, or null set, symbolized ∅.
EXAMPLE Solving a Linear Equation (Fractional Coefficients)
Solve .
Step 1 The LCD of all the fractions in the equation is 6.
The check confirms that the solution set is {−6}.
EXAMPLE Solving a Linear Equation (Fractional Coefficients)
Solve .
Step 1 We clear the parentheses first. Then we clear the fractions.
Step 4 Check to confirm that {−2} is the solution set.
EXAMPLE Solving a Linear Equation (Decimal Coefficients)
Solve 0.1t + 0.05 (20 − t) = 0.09 (20).
The decimals here are expressed as tenths (0.1 and 1.8) and hundredths (0.05). We choose the least exponent on 10 to eliminate the decimal points, which will make all coefficients integers. Here, we multiply by 102—that is, 100.
Step 4 Check to confirm that {16} is the solution set.