Problem 48 - Algebra 1

Question

Evaluate the expression for the given value of x. $$-11+(-2)+11+x ; x=-10$$

Step-by-Step Solution

Answer

The evaluated expression is \(-12\).

Step 1: Substitute the given value into the equation

You are given \(x = -10\). Substitute this value into the equation. This results in \( -11 + (-2) + 11 - 10 \) because we replace \(x\) with \(-10\).

Step 2: Simplify the Equation

Perform the arithmetic operations within the equation. Remember the following basic mathematics rules: for any numbers \(x\) and \(y\), \(x - (-y) = x + y\) and \(-x - y = -(x + y)\). First, perform the addition: \(-11 + (-2) = -13\), then \(-13 + 11 = -2\), and finally, \(-2 - 10 = -12\). Therefore, the final result will be \(-12\).

Key Concepts

Substitution Method

The substitution method is a foundational technique used in mathematics to replace variables with known values to simplify an expression or equation. In the expression given, \(-11 + (-2) + 11 + x\), we're tasked with substituting the value of \(x\) with \(-10\). This method is straightforward: you simply replace every instance of the variable in the expression with the provided numerical value.

By substituting \(x = -10\), the original expression becomes \(-11 + (-2) + 11 - 10\). The substitution step is critical because it transforms the expression from a generalized statement involving variables to a specific numerical expression that we can solve.

This technique is particularly useful in solving equations to find specific values or to modify mathematical models into something more tangible. It streamlines the process of finding answers and helps in verifying if the solution is correct through back-substitution.

Arithmetic Operations

Arithmetic operations are basic mathematical processes that include addition, subtraction, multiplication, and division. When we evaluate an expression like \(-11 + (-2) + 11 - 10\), these operations guide us in processing the numbers step-by-step.

Here is how each operation is applied in the solution:

• First, combine \(-11 + (-2)\) which results in \(-13\). This step involves addition of two negative numbers, where the sum becomes more negative.
• Next, add \(-13 + 11\). This is an addition of a negative and a positive number. When numbers are of opposite signs, you find the difference and keep the sign of the larger absolute value, leading us to \(-2\).
• Finally, subtract \(10\) from \(-2\) to get \(-12\). The subtraction of a positive number from a negative number makes the result more negative, as you move further left on the number line.

Mastering these operations and understanding the rules surrounding them is key to correctly simplifying expressions and solving a wide array of mathematical problems.

Simplifying Expressions

Simplifying an expression involves reducing it to its most basic form to make it easier to work with or understand.

In the evaluated expression \(-11 + (-2) + 11 - 10\), simplifying means combining like terms and performing calculations to reach the simplest form, which in this case is \(-12\).

During this process:

• Combine constants first: Negative and positive values can be combined through addition or subtraction, such as in \(-11 + (-2)\) and then \(-2 + 11\). Simplification often involves combining numbers strategically to reduce complexity.
• The final subtraction \(-2 - 10\) simplifies the expression completely, leaving it as a single value rather than an expression with multiple terms.

Proper simplification ensures that you have the most compact and manageable expression possible.

This concept is vital for algebraic computations, allowing for more accessible manipulation and solution of expressions and equations. Simplification is particularly crucial in higher-level math where expressions become more complex.