Simplifying Expressions and Solving Linear Equations

Expressions and Linear Equations

A constant expression has no variables:
7
¼
0.3

A linear expression has no variables of a power
greater than one. Linear expressions simplify to
Ax + B, where A and B are real numbers.
7x, 3x – 5, X + 3
To simplify a linear expression, we combine the
variable terms and the constant terms.
7x + 3 + 2x – 5 = 9x -2

Examples:
Simplify:
2x – 5 – 3x + 6 17 – 3y + 2y
= 2x – 3x – 5 + 6 = 17 – 1y
= -x + 1 = - y + 17
3x + 9 + 7 – 5x
= 3x – 5x + 9 + 7
= -2x + 16

A linear equation in one variable also has only a
variable raised to the first power:
3x – 5 = 8
3x + 5 = 2x - 6

Finding the value of the variable in a linear
equation is called solving the equation.
The Equation:
Solve x + 3 = 10; X = 7
Is easy enough to solve by inspection, but to
solve more complex equations, we need a
strategy.

The model we will use for
understanding is a scale.
x + 3 = 10
Placing an equal sign between two expressions
indicates they are the same quantity.
The scale is balanced.

To solve the equation, we want to
isolate x on one side of the
equation.
x + 3 = 10 -> x = ____
We do this by “undoing” every operation
that is applied to x on one side of the
equation using inverse operations.

We can easily check our answer: 7 + 3 = 10
Subtraction is the opposite of
(undoes) addition, so we subtract
from both sides of the equation to
get x isolated:
x + 3 = 10
-3 -3
x = 7

Here are a few other equations we
can solve in one-step:
X - 3 = 9
+3 +3
X = 12
3x = 9
3x = 9
3 3
x = 3
x = 9
3
(3)x = (3)9
3
x = 27

Another type of equation you will often see is one requiring
an addition/subtraction and a multiplication/division. Which
step should you do first? It does not matter, but it is usually
less messy to do the addition or subtraction first.
Method 1: Method 2
3x + 6 = 9 3x + 6 = 9
-6 -6 3x + 6 = 9
3x = 3 3 3 3
3x = 3 x + 2 = 3
3 3 - 2 -2
X = 1 x = 1
Same
result
Have to deal with
fractions. Method 1
is often simpler

When we have equations with variables on both sides of a
linear equation. Add or subtract variable expressions to get the
variables on one side.
5x - 9 = 3x – 5
-3x -3x
2x – 9 = -5
+9 +9
2x = 4
2 2
x = 2
Note: I usually
perform an
addition or
subtraction that
leaves a positive
variable
expression, if
possible.
5x - 9 = 3x – 5
-5x -5x
– 9 = -2x - 5
+5 +5
-4 = -2x
-2 -2
2 = x
Same result

Usually, it is best to simplify the expression on
each side of the equation before we start solving.
3(x – 2) + 2x - 3 = 3x – 5
3x – 6 + 2x - 3 = 3x – 5
3x + 2x - 6 - 3 = 3x – 5
5x - 9 = 3x - 5
Now work as in the previous problem

Simplifying Expressions and Solving Linear Equations

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