Simplify and Evaluate Algebraic Expressions

Grade 6 – Mathematics

About Lesson

Simplify:

Simplify means to make it simple. In mathematics, simply or simplification is reducing the expression/fraction/problem in a simpler form. It makes the problem easy with calculations and solving.

- We can simplify fractions by canceling all the common factors from both the numerator and the denominator and writing the fraction in its lowest/simplest form.
- We can simplify mathematical expressions by grouping and combining similar terms. This makes the expression easily understandable and solvable.

Substitution

Substitution means replacing the variables in an algebraic expression with numerical or algebraic values.

Example: Find the value of 5b + 2 when b = 8

Solution: 5b means 5 x b = 5 x 8 = 40

so 5b + 2 = 40 + 2 = 42

Evaluate expressions

To evaluate an algebraic expression, you have to substitute a number for each variable and perform the arithmetic operations. In the example above, the variable x is equal to 6 since 6 + 6 = 12.

If we know the value of our variables, we can replace the variables with their values and then evaluate the expression.

Example: Calculate the following expression for x = 3 and z = 2

$6 z + 4 x = ?$

Solution: Replace x with 3 and z with 2 to evaluate the expression.

Algebraic Manipulation

Algebraic manipulation involves rearranging and substituting for variables to obtain an algebraic expression in a desired form. During this rearrangement, the value of the expression does not change.

Properties of Equality

Any time we add, subtract, multiply, divide, square, square root, etc. to one side, we must do it to the other side to maintain equality.

Example

$begin{align*} x^2 + 18 &= 43 x^2 &= 25 x &= boxed{pm 5} end{align*}$

In the example, we first subtract 18 from the left side to isolate the $x^2$ . However, we also have to subtract 18 from the right side to maintain equality. The right-hand side becomes $43 - 18 = 25$ .

We then square root the left side to get the $x$ by itself. However, we also have to square root the right side to maintain equality.

Cross Multiplication

Cross multiplication is a common method of solving proportions. Essentially, one is multiplying both sides by the denominators of both sides.

Example

$begin{align*} frac{x}{3} &= frac{10}{5} 5 cdot x &= 3 cdot 10 5x &= 30 x &= 6 end{align*}$

The above method of solving is an example of cross-multiplication. During the cross-multiplication, we actually multiplied both sides by $3 cdot 5$ . On the LHS, the $3$ gets divided out (leaving a $5$ for multiplication), and on the RHS, the $5$ gets divided out (leaving a $3$ for multiplication).