Solving Equations
Solving equations is an important skill for problem solving in algebra.
Any replacement for the variable that makes an equation true is called
the solution of the equation. To solve an equation means to find all of
its solutions. If an answer does not make the statement on both sides
of the equals sign true, then there is no solution.
There are two basic principles that we use to solve equations – the
addition and the multiplication principles.
The Addition Principle: For any real numbers a, b, and c, → a = b
Is equivalent to a + c = b + c.
This means we can add anything we want to one side of the equation as
long as we add the same thing to the other side, which balances both
sides of the equation. We isolate x by making the other terms zero that
on the same side of the equals sign as the variable.
x + 5 = -7
-5 -5
x = -12
Then put back in to check!
The Multiplication Principle: For any real numbers a, b, and c, with c ≠
0 → a = b is equivalent to a ∙ c = b ∙ c.
This also balances the equation and we isolate x by making the
coefficient 1.
→
Many times you have to apply both principles as well as the
commutative, associative, and distributive laws to write equivalent
expressions.
If there is only one variable, always apply the addition principle before
the multiplication principle.
5 + 3x = 17
-5
-5
3x = 12
3 3
x=4
Combining Like Terms
If like terms appear on the same side of the equation, we combine like
terms first then solve using addition/multiplication principles. If like
variables appear on both sides of the equals sign, use the addition
principle to rewrite all like terms on one side.
3x + 4x = -14
7x =-14
x = -2
2x – 4= -3x + 1
3x
3x
5x -4 = 1
+4 +4
5x = 5 → x = 1