# Simplifying Expressions to Solve Equations

Like terms are terms with the same variable(s) and the
same power.

A good way to familiarize yourself with like terms is to compare and contrast :

Like terms | Unlike terms |

Ex.1

Simplify by combining like terms :

1. 7x + 3x *A good way to look at the variables is making them a unit of
measure. For

example, make the x a dollar and read it this way, “7 dollars + 3 dollars.” The
answer is

obviously 10 dollars. It’s the same concept with variable terms. So 7x + 3x =
10x.

2. -5a + 2b -a + 3b *Because of the commutative property of addition (and
remember that

subtraction is a form of addition), I can rearrange these terms if necessary. So
putting my

like terms together gives me: -5a -a + 2b + 3b (notice that only the order
changed , not

the signs). Now combining like terms gives me an answer of -6a + 5b.

Ex. 2

Use the distributive property to simplify:

1. 3(x + 2) *Normally in the order of operations, we would work operations
inside the

parenthesis first , but in this case, x and 2 are not like terms and can not be
combined. The

only way to simplify this expression is by using the distributive property. So
we’ll

distribute the 3 this way: 3(x) + 3(2). Note that 3(x) is 3x and 3(2) is 6 so
the answer is

3x + 6.

2. -2(y + 8) *Again, we would do the order of operations if possible, but this
is not the case.

So we’ll use the distributive property to rewrite: -2(y) + -2(8) and we’ll get
-2y -16 for

our answer. Remember that since subtraction is the opposite of addition, this
could also

be written as -2y + -16 or -16 -2y.

3. 5(2a + 3) + 2(a -5) *The only difference with this
expression is that we have two sets of

parentheses. Distribute the 5 through the first set of parenthesis and the 2
through the

second set to get: 5(2a) + 3 + 2(a) + 2(-5). This will leave us with 10a + 3 +
2a -10.

Unlike the others, we have like terms in this expression that can be combined
(the 10a

and the 2a, etc.). You can rearrange if necessary and get: 10a + 2a + 3 -10.
After

combining the like terms we get 12a -7.

Steps for solving linear equations :

1) Simplify the expression on each side of the equal sign ( clear fractions ,
clear parentheses,

combine like terms, etc.).

2) Get variable terms on one side of the equation and constants on the other.

3) Isolate the variable and solve the equation .

4) Check the solution .

Ex. 3

Solve each equation:

1. 3x + 6 -5x = 10 *This equation is unlike others that we have solved because
we have two

variable terms in the equation. Looking at the steps for solving equations, we
see that the

first step is to simplify the expressions on each side of the equal sign. I
notice on the left

side, I have like terms that can be combined: -2x + 6 = 10. Now I move to step 2
and get

my variable terms (I only have one) on one side and constants on the other: -2x
= 4

(remember to subtract 6 from both sides of the equation). Now I’ll isolate the
variable by

dividing both sides of the equation by -2 and get x = -2. To check your
solution, always

go back to the original equation: . Using our
order of

operations: -6 + 6 + 10 is indeed 10.

2. 2(4y -1) + 3(y + 2) = 11 *This time, before combining like terms we have to
get rid of

the parentheses. We know that we do this using the distributive property:

8y -2 + 3y + 6 = 11. Now combining like terms gives us: 11y + 4 = 11. Now we get
the

constants on the right side: 11y = 7. Isolate the variable to get
.

Ex. 4

Solve each equation:

1. 3(2x -5) + 7 = 4(x + 9) *This equation is not too different from the example
above but it

involves variable terms on each side of the equal sign. We’ll get to that in a
minute.

First, get rid of the parentheses by using the distributive property: 6x -15 + 7
= 4x + 36.

On the left side of the equal sign, we have some like terms to combine: 6x -8 =
4x + 36.

At this point, we can’t add like terms (6x & 4x) until they are on the same side
of the

equal sign. We’ll use the same procedure that we learned in section 2.1:

This step yields: | |

2x -8 = 36. Now this looks just like the equations in 2.2. We’ll get rid of the
-8 by

adding it to both sides: 2x = 44. Finally, we’ll divide by 2 on both sides and
get x = 22.

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