A linear equation in one variable is an equation which can
be written in the form:
ax + b = c
for a, b, and c real numbers with a ≠0.
Linear equations in one variable:
2x + 3 = 11
2(x - 1) = 8 can be rewritten 2x + (- 2) = 8.
can be rewritten
Not linear equations in one variable:
A solution of a linear equation in one variable is a real
number
which, when substituted for the variable in the equation, makes
the equation true.
Example: Is 3 a solution of 2x + 3 = 11?
 |
Original equation |
| Substitute 3 for x. |
| False equation |
3 is not a solution of 2x + 3 = 11.
Example: Is 4 a solution of 2x + 3 = 11?
| 2x + 3 = 11 |
Original equation |
| 2(4) + 3 = 11 |
Substitute 4 for x. |
| 8 + 3 = 11 |
True equation |
4 is a solution of 2x + 3 = 11.
That is, the same number can be added to or subtracted
from
each side of an equation without changing the solution of the
equation.
Use these properties to solve linear equations.
Example: Solve x - 5 = 12.
 |
Original equation |
The solution is preserved when 5 is
added to both sides of the equation. |
| 17 is the solution. |
| Check the answer. |
That is, an equation can be multiplied or divided by the
same
non zero real number without changing the solution of the
equation.
Example: Solve 2x + 7 = 19.
 |
Original equation |
 |
The solution is preserved when 7 is
subtracted from both sides. |
 |
Simplify both sides. |
 |
The solution is preserved when each side
is multiplied by 1/2. |
 |
6 is the solution |
 |
Check the answer. |
To solve a linear equation in one variable:
1. Simplify both sides of the equation.
2. Use the addition and subtraction properties to get all variable terms
on the left-hand side and all constant terms on the right-hand side.
3. Simplify both sides of the equation.
4. Divide both sides of the equation by the coefficient of the variable. |
Example: Solve x + 1 = 3(x - 5).
| x + 1 = 3(x - 5) |
Original equation |
| x + 1 = 3x - 15 |
Simplify right-hand side. |
| x = 3x - 16 |
Subtract 1 from both sides. |
| - 2x = - 16 |
Subtract 3x from both sides. |
| x = 8 |
Divide both sides by -2. |
The solution is 8.
Check the solution:(8) + 1 = 3((8) - 5)→ 9 = 3(3) True
Example: Solve 3(x + 5) + 4 = 1 – 2(x + 6).
| 3(x + 5) + 4 = 1 – 2(x + 6) |
Original equation |
| 3x + 15 + 4 = 1 – 2x – 12 |
Simplify. |
| 3x + 19 = – 2x – 11 |
Simplify. |
| 3x = – 2x – 30 |
Subtract 19. |
| 5x = – 30 |
Add 2x. |
| x = - 6 |
Divide by 5. |
The solution is - 6.
| 3(– 6 + 5) + 4 = 1 – 2(– 6 + 6) |
Check. |
| 3(– 1) + 4 = 1 – 2(0) |
|
| - 3 + 4 = 1 |
True |
Equations with fractions can be simplified by multiplying
both
sides by a common denominator .
Example:
Solve  |
The lowest common denominator
of all fractions in the equation is 6 |
 |
Multiply by 6. |
| 3x + 4 = 2x + 8 |
Simplify. |
| 3x = 2x + 4 |
Subtract 4. |
| x = 4 |
Subtract 2x. |
 |
Check. |
 |
|
 |
True |
Alice has a coin purse containing $5.40 in dimes and
quarters.
There are 24 coins all together. How many dimes are in the coin
purse?
Let the number of dimes in the coin purse = d.
Then the number of quarters = 24 - d.
 |
Linear equation |
| 10d + 600 - 25d = 540 |
Simplify left-hand side. |
| 10d - 25d = - 60 |
Subtract 600. |
| -15d = - 60 |
Simplify right-hand side. |
| d = 4 |
Divide by -15. |
There are 4 dimes in Alice’s coin purse.
The sum of three consecutive integers is 54. What are the
three integers?
Three consecutive integers can be represented as
n, n + 1, n + 2.
 |
Linear equation |
| 3n + 3 = 54 |
Simplify left-hand side. |
| 3n = 51 |
Subtract 3. |
| n = 17 |
Divide by 3. |
The three consecutive integers are 17, 18, and 19.
17 + 18 + 19 = 54. Check
