Problem 41 - Algebra 1: Concepts and Skills

Question

Evaluate the variable expression when c 4 and d 5. $$ \left(d^{2}\right)+c $$

Step-by-Step Solution

Answer

The evaluated value of the expression \(d^{2} + c\) for d = 5 and c = 4 is 29.

Step 1: Identify the variables and the expression

Here the expression is \(d^{2} + c\). The variables are 'd', and 'c'. They are given the values d = 5 and c = 4.

Step 2: Substitution

Substitute the given values into the variables 'd' and 'c' in the expression. So, it becomes \(5^{2} + 4\).

Step 3: Evaluate according to the BODMAS rule

According to the BODMAS rule, we have to calculate the power before the addition. So, \(5^{2}\) is 25. Now the expression becomes \(25 + 4\).

Step 4: Perform Addition

Perform the addition operation. So, \(25 + 4 = 29\).

Key Concepts

BODMAS rule

When evaluating mathematical expressions, the BODMAS rule (also called PEMDAS in some places) helps us to know the correct order to follow. BODMAS stands for:

• Brackets
• Orders (powers and roots)
• Division
• Multiplication
• Addition
• Subtraction

This order tells us that operations inside brackets get solved first. Then, we take care of powers (or exponents) and roots. Next, we deal with division and multiplication, from left to right. Lastly, addition and subtraction are carried out, also from left to right. The BODMAS rule can be helpful in organizing and simplifying the way we look at calculations, ensuring we derive the correct answer every time. In our example, we first evaluated the exponent in the expression, with the power calculation \(5^2\) being performed before performing the addition with the rest of the expression.

Substitution in algebra

Substitution in algebra is a method used to replace variables in an expression or equation with given values. This process helps to simplify the expression and make calculations possible.

In the context of our example, we start with the expression \(d^2 + c\). Here, the variables are 'd' and 'c'. We are given specific values for these variables: \(d = 5\) and \(c = 4\). To proceed with the evaluation, we substitute these values in place of the variables within the expression. So the expression \(d^2 + c\) becomes \(5^2 + 4\).

Substitution is a crucial step in solving algebraic expressions, as it allows us to work with numerical values instead of symbols, making it easier to apply arithmetic operations and continue with evaluation.

Variable expressions

Variable expressions are mathematical statements that contain variables like \(c\), \(d\), or \(x\). These variables represent numbers that can take on different values, which means the expression itself can be evaluated in various ways depending on these values.

In the context of our exercise, the expression given is \(d^2 + c\). Here, 'd' and 'c' are variables that represent specific numbers, which have been provided as \(d = 5\) and \(c = 4\). By using substitution, as previously discussed, we can assign these specific numbers to the variables within the expression.

This results in a numerical expression that can be easily solved using arithmetic operations. By understanding variable expressions, students can explore how changing the values of variables impacts the overall expression and the final evaluation. This is a fundamental skill that underpins much of algebra and higher-level math.