Evaluate $10x - 2y$ When $x=2$ And $y=3$

Nov 11, 2025 by ADMIN 41 views

Alright, let's break down how to evaluate the multi-variable expression $10x - 2y$ when $x = 2$ and $y = 3$ . This is a fundamental concept in algebra, and mastering it will help you tackle more complex problems down the road. The key here is substitution: we replace the variables (which are $x$ and $y$ in this case) with their given values and then perform the arithmetic operations according to the order of operations (PEMDAS/BODMAS). Understanding the order of operations is crucial for ensuring we get the correct answer. This means handling parentheses/brackets first, then exponents/orders, then multiplication and division (from left to right), and finally addition and subtraction (from left to right). So, let's dive in and see how this works step by step!

Step-by-Step Evaluation

To evaluate the expression, we will follow these steps:

Substitute the values of $x$ and $y$ into the expression.
Perform the multiplication.
Perform the subtraction.

1. Substitute the values

We are given that $x = 2$ and $y = 3$ . We substitute these values into the expression $10x - 2y$ . This gives us:

$10(2) - 2(3)$

2. Perform the multiplication

Next, we perform the multiplication operations. We have two multiplication operations here: $10 * 2$ and $2 * 3$ .

$10 * 2 = 20$

$2 * 3 = 6$

So our expression now looks like:

$20 - 6$

3. Perform the subtraction

Finally, we perform the subtraction operation:

$20 - 6 = 14$

Therefore, the value of the expression $10x - 2y$ when $x = 2$ and $y = 3$ is $14$ .

Detailed Explanation

Let's go through each step in more detail to ensure we understand the underlying concepts. When we substitute $x = 2$ and $y = 3$ into the expression $10x - 2y$ , we're essentially replacing the symbols $x$ and $y$ with the numbers they represent in this specific problem. This is a fundamental concept in algebra: variables hold placeholders for values, and substitution allows us to evaluate the expression for particular instances of those values. Think of it like filling in blanks in a sentence; the variables are the blanks, and the given values are the words we use to fill them. Once we've substituted the values, the expression becomes a purely arithmetic one, involving only numbers and operations. Now, following the order of operations, we tackle the multiplication first. Multiplying $10$ by $2$ gives us $20$ , and multiplying $2$ by $3$ gives us $6$ . This simplifies the expression to $20 - 6$ . Finally, we perform the subtraction. Subtracting $6$ from $20$ yields $14$ . So, the final value of the expression is $14$ . This result is a single number because we have fully evaluated the expression for the given values of $x$ and $y$ . This process illustrates how algebraic expressions can be used to represent relationships between variables, and how substitution allows us to find specific values based on those relationships.

Common Mistakes to Avoid

When evaluating expressions like this, it's easy to make a few common mistakes. Here are some pitfalls to watch out for:

Forgetting the order of operations: Always remember PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Make sure you perform multiplication before addition or subtraction.
Incorrect substitution: Double-check that you're substituting the correct values for the correct variables. It's easy to mix them up, especially when the expression has multiple variables.
Arithmetic errors: Simple arithmetic mistakes can throw off your entire calculation. Take your time and double-check your work, especially during multiplication and subtraction.
Misinterpreting the expression: Make sure you understand exactly what the expression is asking you to do. For example, $10x$ means $10$ times $x$ , not $10 + x$ or some other operation.

Avoiding these common mistakes will help you evaluate expressions accurately and confidently. Always double-check your work and pay attention to the details.

Practice Problems

To solidify your understanding, let's work through a few more practice problems:

Evaluate $5a + 3b$ when $a = 4$ and $b = 1$ .
Evaluate $x^2 - 2y$ when $x = 3$ and $y = 2$ .
Evaluate $4m - n/2$ when $m = 5$ and $n = 8$ .

Solutions to Practice Problems

$5a + 3b$ when $a = 4$ and $b = 1$ :

$5(4) + 3(1) = 20 + 3 = 23$
$x^2 - 2y$ when $x = 3$ and $y = 2$ :

$3^2 - 2(2) = 9 - 4 = 5$
$4m - n/2$ when $m = 5$ and $n = 8$ :

$4(5) - 8/2 = 20 - 4 = 16$

Real-World Applications

Evaluating multi-variable expressions isn't just a theoretical exercise; it has many real-world applications. For example:

Physics: In physics, you might use expressions to calculate the force on an object, the energy of a system, or the velocity of a projectile. These expressions often involve multiple variables that represent different physical quantities.
Engineering: Engineers use multi-variable expressions to design structures, analyze circuits, and model fluid flow. These expressions help them predict how systems will behave under different conditions.
Economics: Economists use expressions to model economic phenomena, such as supply and demand, inflation, and GDP growth. These expressions can help them understand how different factors influence the economy.
Computer Science: In computer science, multi-variable expressions are used in algorithms and data structures. For example, you might use an expression to calculate the time complexity of an algorithm or the memory usage of a data structure.

By understanding how to evaluate multi-variable expressions, you'll be better equipped to solve problems in these and other fields. The ability to manipulate and evaluate these expressions is a fundamental skill that will serve you well in many areas of life.

Conclusion

In summary, evaluating the multi-variable expression $10x - 2y$ when $x = 2$ and $y = 3$ involves substituting the given values into the expression and performing the arithmetic operations according to the order of operations. By following these steps carefully, we arrive at the solution $14$ . Remember to avoid common mistakes and practice regularly to improve your skills. Whether you're working on math problems, physics equations, or real-world applications, mastering the art of evaluating expressions will be a valuable asset. Keep practicing, and you'll become more confident and proficient in your mathematical abilities!