How to Find the Value of an Expression with Variables

📚 Understanding Variable Expressions

In mathematics, an expression is a combination of numbers, variables, and operation symbols. When we talk about finding the 'value' of an expression with variables, we mean figuring out what number the expression equals when we replace the variables with specific numbers. This is a fundamental concept in algebra and is crucial for solving equations and modeling real-world scenarios.

📜 A Brief History

The use of variables in mathematical expressions dates back to ancient civilizations. Early forms of algebra were developed by the Babylonians and Egyptians, who used symbols to represent unknown quantities. However, the modern notation we use today, with letters like 'x' and 'y' representing variables, became more standardized in the 16th and 17th centuries, thanks to mathematicians like François Viète and René Descartes.

🔑 Key Principles

• 🔢 Substitution: The first step is replacing each variable in the expression with its given value. For instance, if you have the expression $3x + 2y$ and $x = 5$ and $y = 1$, you substitute to get $3(5) + 2(1)$.
• 🧮 Order of Operations: After substituting, follow the order of operations (often remembered by the acronym PEMDAS/BODMAS): Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right).
• ➕ Simplification: Perform the arithmetic operations to simplify the expression into a single numerical value. In the example above, $3(5) + 2(1)$ becomes $15 + 2$, which simplifies to $17$.

➗ Real-World Examples

Let's go through some practical examples to illustrate how to find the value of an expression with variables:

1. Example 1: Suppose you have the expression $2a - b + 4c$, where $a = 3$, $b = 2$, and $c = 1$.
   • Substitute: $2(3) - 2 + 4(1)$
   • Multiply: $6 - 2 + 4$
   • Add/Subtract: $4 + 4 = 8$
   • Therefore, the value of the expression is $8$.
2. Example 2: Consider the expression $x^2 + 3y - z$, where $x = 4$, $y = -1$, and $z = 5$.
   • Substitute: $(4)^2 + 3(-1) - 5$
   • Exponent: $16 + 3(-1) - 5$
   • Multiply: $16 - 3 - 5$
   • Add/Subtract: $13 - 5 = 8$
   • Therefore, the value of the expression is $8$.
3. Example 3: Calculate the value of $\frac{p + q}{2} - r$, given $p = 7$, $q = 5$, and $r = 2$.
   • Substitute: $\frac{7 + 5}{2} - 2$
   • Add (in numerator): $\frac{12}{2} - 2$
   • Divide: $6 - 2$
   • Subtract: $4$
   • Therefore, the value of the expression is $4$.

📝 Practice Quiz

Test your understanding with these practice problems. Evaluate each expression for the given values:

1. $5x + 2$, where $x = 3$
2. $y^2 - 4$, where $y = 6$
3. $3a - 2b + c$, where $a = 2$, $b = 1$, $c = 5$
4. $\frac{m}{2} + n$, where $m = 10$, $n = 4$
5. $p^2 - q$, where $p = 5$, $q = 8$
6. $4(r - s)$, where $r = 7$, $s = 3$
7. $x + y - z$, where $x = 9$, $y = 2$, $z = 6$

💡 Solutions to Practice Quiz

1. $5(3) + 2 = 15 + 2 = 17$
2. $(6)^2 - 4 = 36 - 4 = 32$
3. $3(2) - 2(1) + 5 = 6 - 2 + 5 = 9$
4. $\frac{10}{2} + 4 = 5 + 4 = 9$
5. $(5)^2 - 8 = 25 - 8 = 17$
6. $4(7 - 3) = 4(4) = 16$
7. $9 + 2 - 6 = 11 - 6 = 5$

✅ Conclusion

Finding the value of an expression with variables involves substituting given values for variables and then simplifying the expression using the order of operations. With practice, this becomes a straightforward process, essential for more advanced algebraic concepts. Keep practicing, and you'll master it in no time!