Simplifying equations is often the first step to solving algebra equations. Some methods that can be used to simplify an equation are Combine Like Terms, Multiplication & Division of Terms, Removal of Brackets (Distributive Property) and Cross Multiplication.

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Isolate A Variable

Algebraic Expressions

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The following diagrams show how to combine like terms to simplify an expression. Scroll down the page for more examples and solutions.

Like terms are terms that have the same variable part, i.e. they only differ in their coefficients. Combining like terms is very often required in the process of simplifying equations.

For examples:

2*x* and –5*x* are like terms.

*a* and are like terms.

6*x* and 5*y* are **unlike** terms.

Like terms can be added or subtracted from one another.

For example:

*a* + *a* = 2 × *a* = 2*a* (We usually write 2 × *a* as 2*a*)

2*a* + 4*a* = 6*a*

*a* + *a* + *a* = 3*a*

2*a* + 4 (Unlike terms cannot be simplified)

4*a* + 3*b* (Unlike terms cannot be simplified)

6*a* – 3*a* = 3*a*

8*b* – 8*b* = 0

5*a* – 3 (Unlike terms cannot be simplified)

6*a* – 4*b* (Unlike terms cannot be simplified)

Example 1:

Simplify: 8*xy* – 5*yx* = 1

Solution:

Step 1: 5*yx* is the same as 5*xy* using the commutative property

Step 2: Since the right side is already simple, we can work on the left side expression

8*xy* – 5*yx* = 8*xy* – 5*xy* = 3*xy*

Putting back the left side and right side of the equation:

3*xy* = 1

Answer: 3*xy* = 1

Example 2:

Simplify: 7*a* + 5*b* – 6*b* + 8*a* + 2*b* = 0

Solution:

Step 1: Group together the like terms:

7*a* + 5*b* – 6*b* + 8*a* + 2*b* = 0

(7*a* + 8*a*) + (5*b* – 6*b* + 2*b*) = 0

Step 2: Then simplify:

15*a* + *b* = 0

Answer: 15*a* + *b* = 0

The coefficients and variables of terms can be multiplied or divided together in the process of simplifying equations.

For examples:

3 × 4*b* = 3 × 4 × *b* = 12*b*

5*a* × 3*a* = 5 × *a* × 3 × *a* = 5 × 3 × *a* × *a* = 15*a*^{2} (using exponents)

Be careful!

*a* × *a* = *a*^{2}< br>
*a *+ *a* = 2*a*

Example 1:

Simplify: 4*a* × 5*a* ÷ 2*a* = 60

Solution:

Step 1: Perform the multiplication and division

Step 2: Isolate *a*

Answer: *a* = 6

Sometimes removing brackets (parenthesis) allows us to simplify the expression. Brackets can be removed by using the distributive property. This is often useful in simplifying equations.

For example:

3(*a* – 3) + 4 = 3 × *a* + 3 × (-3) + 4 = 3*a* – 9 + 4 = 3*a* – 5

5 – 6 (*b* + 1) = 5 + ( – 6 ) × *b* + (– 6) × 1 = 5 – 6*b* – 6 = – 6*b* – 1

Example 1:

Simplify: 5(*a* – 4) + 3 = 8

Solution:

Step 1: Remove the brackets

5*a* – 20 + 3 = 8

Step 2: Isolate variable *a *

5*a* = 8 – 3 + 20

5*a* = 25

Answer: *a* = 5

Cross multiplication allows you to remove denominators from fractions in an equation. Note that this technique applies only towards simplifying equations, not to simplifying expressions.

For example, if you have the equation:

then you can multiply the numerator of one fraction with the denominator of the other fraction (across the = sign) as shown:

to obtain the equation

(2 × 6) = *a* × 3

Example 1:

Simplify:

Solution:

Step 1:Cross Multiply

4 × *a* = 8 × 5

4*a* = 40

Step 2: Isolate variable *a*

Answer:* a* = 10

Have a look at the following video for more examples on simplifying equations.

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