Solving Linear Equations

In this note we will review solving equations that are either linear or lead to linear equations.

Example 1

Solve 3x - 2 = 12 and check your answer.

Solution .

Add 2 to both sides we get

Multiply both sides by 1/3 we have

Substituting this into the original we have

Thus the solution is x = 14/3.

Example 2

Solve the equation

Solution .

Expand both sides

Add to both sides

Add to both sides

Add -9 to both sides

Multiply both sides of by 1/4

Putting this value into the original equation we get

x = -5/4 is the solution.

Example 3

Solve

Solution .

Note that because of the denominator, x cannot equal -1.

Simplify the left side. Since it is equivalent to (x+1)/(x+1), we have

Since this is nonsense, the equation has no solution.

Example 4

Solve

Solution .

Factor the denominator.

Because of the x-3 and x+2 in the denominators, we know that x cannot be 3 and x cannot be -2.

To clear the fraction multiply both sides by the LCD of (x-3)(x+2) .

Use the distributive property on the left side of the equation and simplify.

Add -1 to both sides

Multiply both sides by 1/3

This meets the restrictions noted earlier. Putting x = 2 into the original equation yields

Thus, x = 2 is the solution.