Last modified on December 18th, 2023

chapter outline

 

Evaluating Algebraic Expressions

Evaluating an algebraic expression means finding the value of the given expression when a variable is substituted with its value and then performing the arithmetic operations to get the result. 

Let us now evaluate the expression x + 2 for a) x = 15, b) x = 12

a) Substituting the given value of x  = 15  in the given expression, we get,

15 + 2 = 17

b) Substituting the given value of x = 12 in the same expression, we get,

12 + 2 = 14

Now let us substitute x = 7 in the expression 2x – 5, we get,

2(7) – 5 = 14 – 5 = 9

After substitution, we simplify the expression using the order of operations (PEMDAS) rule. 

Let us evaluate an algebraic expression ${4s^{2}-s+3}$, for s = 5

First, we will substitute ‘s’ with 5 in the given expression.

= ${4\left( 5\right) ^{2}-\left( 5\right) +3}$

The expression involves multiplication, an exponent, subtraction, and addition. According to the order of operations (PEMDAS), we simplify exponents and perform multiplication, addition, and subtraction. 

Thus, we get ${4\times 25-5+3}$

Now, by multiplying, we get 100 – 5 + 3

By adding, we get 103 – 5

And finally, by subtracting, we get 98

Thus, the given expression ${4s^{2}-s+3}$ is evaluated to 98, when s = 5.

For More than One Variable

Now, let us evaluate the expression ${a^{2}+3bc}$, when a = 2, b = 6, c = 1

First, substitute a with 2, b with 6, and c with 1 in the given expression.

We get ${\left( 2\right) ^{2}+3\left( 6\right) \left( 1\right)}$

Here, the expression involves addition, multiplication, and an exponent. According to the order of operations, we solve the exponent first, then multiply and add to simplify further.

Thus, we get ${4+3\left( 6\right) \left( 1\right)}$

Now, by multiplying, we get 4 + 18

By adding, we get 22

Thus, the given expression ${a^{2}+3bc}$ is evaluated to 22, when a = 2, b = 6, c = 1.

With Fractions

In the case of algebraic expressions with fractions, we always evaluate and simplify the numerator and the denominator separately. Then, we remove the common factor from both to get the required value.

Let us evaluate the expression ${\dfrac{a+3bc}{a^{2}}}$, when a = 2, b = 1, c = 6

First, substitute 2 for a, 1 for b, and 6 for c in the given expression.

We get ${\dfrac{\left( 2\right) +3\left( 1\right) \left( 6\right) }{\left( 2\right) ^{2}}}$

Here, the expression involves addition and multiplication in the numerator and an exponent in the denominator. 

According to PEMDAS, we multiply and add in the numerator and solve the exponent in the denominator. Removing the common factor and rewriting the remaining terms to simplify further, we get,

= ${\dfrac{2+18}{4}}$

= ${\dfrac{20}{4}}$

= 5

Thus, the given expression ${\dfrac{a+3bc}{a^{2}}}$ is evaluated to 5, when a = 2, b = 1, c = 6.

Solved Examples

Evaluate: 4x + (12 – y) – 3z, when x = 2, y = 6, and z = -5

Solution:

To evaluate the given expression 4x + (12 – y) – 3z, substituting the given values 2 for x, 6 for y, and -5 for z, we get,
4(2) + (12 – 6) – 3(-5)
Now, solving using the order of operations, we get,
8 + 6 + 15
= 29
Thus, 4x + (12 – y) – 3z equals to 29, when x = 2, y = 6, and z = -5.

Evaluate the following algebraic expressions.
a) ${\dfrac{3\left( a+c^{2}\right) }{12}}$, when a = -9 and c = 5
b) ${7x^{2}-11x+13}$, when x = 5

Solution:

a) To evaluate the given expression ${\dfrac{3\left( a+c^{2}\right) }{12}}$, substituting the given values -9 for a and 5 for c, we get,
${\dfrac{3\left\{ \left( -9\right) +\left( 5\right) ^{2}\right\} }{12}}$
Now, solving using the order of operations, we get,
${\dfrac{3\left\{ -9+25\right\} }{12}}$
= ${\dfrac{3\times 16}{12}}$
= 4
Thus, ${\dfrac{3\left( a+c^{2}\right) }{12}}$ equals to 4, when a = -9 and c = 5.
b) To evaluate the given expression ${7x^{2}-11x+13}$, substituting the given value 5 for x, we get,
${7\left( 5\right) ^{2}-11\left( 5\right) +13}$
Now, solving using the order of operations, we get,
${7\times 25-55+13}$
= 175 – 55 + 13
= 188 – 55
= 133
Thus, ${7x^{2}-11x+13}$ equals to 133, when x = 5.

Evaluate the algebraic expression: 5x + 2y – 4 
a) When x = 3, y = 11
b) When x = -5, y = 7

Solution:

The given expression is 5x + 2y – 4 
a) Here, by substituting the values 3 for x and 11 for y, we get,
5(3) + 2(11) – 4
Now, solving using the order of operations, we get,
15 + 22 – 4
= 37 – 4
= 33
Thus,  5x + 2y – 4 = 133, when x = 3, y = 11.
b) Here, by substituting the values -5 for x and 7 for y, we get,
5(-5) + 2(7) – 4
Now, solving using the order of operations, we get,
-25 + 14 – 4
= 14 – 29
= -15
Thus,  5x + 2y – 4 = -15, when x = -5, y = 7.

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