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Thinking
Mathematically
Algebra: Equations and Inequalities
6.2 Solving Linear Equations
“Solving” Linear Equations
Two algebraic expressions connected with an
“equal” sign is called an “equation.” When there
are no exponents for the variables (other than one)
the equation is called “linear.”
“Solving” an equation means finding all of the
numbers for the variables that make the equal sign
true. For equations with only one variable, the
“solution” is the set of all of those numbers.
The Addition Property of Equality
The same real number (or algebraic
expression) may be added to both sides of
an equation without changing the solution
set. This can be expressed symbolically as
follows:
If a = b, then a + c = b + c.
The Subtraction Property of Equality
The same real number (or algebraic
expression) may be subtracted from both
sides of an equation without changing the
solution set.
If a = b, then a – c = b – c.
“Solving”Linear Equations
The rules of equality can be used to “solve” a linear
equation. Any number or variable can be added or
subtracted from both sides. The “goal” is to isolate
the variable on one side of the equal sign.
Exercise Set 6.2 #3
x + 5 = -12
The Multiplication Property of
Equality
The same nonzero real number (or algebraic
expression) may multiply both sides of an
equation without changing the solution set.
If a = b and c ≠ 0, then ac = bc.
The Division Property of Equality
Both sides of an equation may be divided by
the same nonzero real number (or algebraic
expression) without changing the solutions
set.
If a = b and c ≠ 0, then a/c = b/c.
“Solving”Linear Equations
The rules of equality can be used to “solve” a linear
equation. Both sides of an equation may be
multiplied or divided by any (non-zero) number or
variable. The “goal” is to isolate the variable on
one side of the equal sign.
Exercise Set 6.2 #7
5x = 45
Solving a Linear Equation
1. Simplify the algebraic expression on each side.
2. Collect all the variable terms on one side and all
the constant terms on the other side.
3. Isolate the variable and solve.
4. Check the proposed solution in the original
equation.
Exercise Set 6.2 #19, #23, #41
14 – 5x = -41
5x – (2x – 10) = 35
6 = -4(1 – x) + 3(x + 1)
Misc.
• Solving linear equation with fractions
Exercise Set 6.2 #47
x x 5
 
3 2 6
•Special cases: no solution, all real numbers
Exercise Set 6.2 #75, 81
2(x + 4) = 4x + 5 – 2x + 3
4(x + 2) + 1 = 7x – 3(x – 2)
Thinking
Mathematically
Algebra: Equations and Inequalities
6.2 Solving Linear Equations