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1.2 Linear Equations
and
Rational Equations
Terms Involving Equations
3x - 1 = 2
Left Side
Right Side
An equation consists of two algebraic expressions joined by an equal sign.
3x – 1 = 2
3x = 3
x=1
1 is a solution or root of the equation.
If you have a SOLUTION to an equation, that is the value for
the variable(s) that make the equation “true”.
Definition of a Linear Equation
• A linear equation in one variable x is an
equation that can be written in the form
• ax + b = 0
• where a and b are real numbers and a ≠ 0.
Generating Equivalent Equations
An equation can be transformed into an equivalent equation by one or more of the following
operations.
1. SIMPLIFY each side by
removing grouping symbols and
combining like terms.
3(x - 6) = 6x - x
3x - 18 = 5x
2. ADD (or subtract) the same
real number or variable
expression on both sides of the
equation to get the variable on
one side, and constants on the
other.
3x - 18 = 5x
3x - 18 - 3x = 5x - 3x
-18 = 2x
3. DIVIDE (or multiply)on both
sides of the equation by the same
nonzero quantity.
-18 = 2x
-9 = x
4. Write your final solution (for
a linear equation with one
variable) with the variable on the
left.
-9 = x
x = -9
Subtract 3x from both
sides of the equation.
Divide both sides of the
equation by 2.
Solving a Linear Equation
• OPTIONAL: multiply through by LCD or
multiple of 10 to clear fractions or decimals
• SIMPLIFY the algebraic expression on each
side.
• ADD to collect all the variable terms on one
side and all the constant terms on the other
side.
• DIVIDE by the coefficient of the variable
to isolate the variable.
• Check the proposed solution in the original
equation.
Ex.
Solve the equation: 2(x - 3) - 17 = 13 - 3(x + 2).
Solution
2(x - 3) – 17 = 13 – 3(x + 2)
This is the given equation.
Step 1
SIMPLIFY the
algebraic expression on each
side.
Step 2
ADD to collect
variable terms on one side and
constant
terms on the other
side.
Step 3
DIVIDE by the
coefficient of the variable to
isolate the variable and solve.
Con’t.
Step 4
Check the proposed solution in the original equation.
Substitute 6 for x in the original equation.
?
2(x - 3) - 17 = 13 - 3(x + 2)
This is the original equation.
Substitute 6 for x.
The solution set is {____}.
Types of Equations
• Identity:An equation that is true for all real
numbers. (0 = 0, all real numbers)
• Conditional: An equation that is true for at
least one real number. (x = 0, or any
constant)
• Inconsistent: An equation that is not true
for any real number. (0 = 5, NO
SOLUTION)
Example
Determine whether the equation 3(x - 1) = 3x + 5 is an identity, a conditional
equation, or an inconsistent equation.
Solution
To find out, solve the equation.
3(x – 1) = 3x + 5
This equation is _________________.
Do p 104 # 27 & 49 in class.