Problem 80 - Elementary and Intermediate Algebra

Question

Evaluate each expression, for \(x=3, y=-2,\) and \(z=-4\) See Example 10. $$ -y-5 x $$

Step-by-Step Solution

Answer

The expression evaluates to -13.

Step 1: Identify the Expression

The given expression is \[-y - 5x\]where we need to substitute the given values of x and y.

Step 2: Substitute the Values

Substitute the values of \(x = 3\) and \(y = -2\) into the expression:\[-(-2) - 5(3)\].

Step 3: Simplify the Expression

Simplify the expression by first resolving the negative signs and multiplication:\[2 - 5 \times 3\].

Step 4: Perform the Multiplication

Multiply \(5\) by \(3\):\[2 - 15\].

Step 5: Solve the Final Expression

Subtract \(15\) from \(2\):\[2 - 15 = -13\].

Key Concepts

Substitution Method

The substitution method is a fundamental technique used in mathematics to evaluate expressions by replacing variables with numbers. In this approach, we take the given values for variables and systematically substitute them into the expression.
Substitution is critical when dealing with algebraic expressions as it allows for numerical evaluation.

In our exercise example, we are given the expression \(-y - 5x\) and need to substitute the values \(x = 3\) and \(y = -2\):

• First, replace \(y\) with \(-2\) to get \(-(-2)\).
• Then, substitute \(x = 3\) into \(5x\) resulting in \(5(3)\).

By doing this, the expression morphs into a numerical one, paving the way for straightforward calculation. This process underscores the power of substitution in turning abstract symbols into tangible numbers allowing us to solve and analyze expressions easily.

Arithmetic Operations

Arithmetic operations refer to the basic processes used to combine numbers and include addition, subtraction, multiplication, and division. These are the fundamental building blocks of mathematics that we rely on to carry out calculations.
When performing arithmetic operations, we must carefully consider the order of operations. Known by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division (left-to-right), Addition and Subtraction (left-to-right)), it guides us in executing calculations correctly. In our exercise scenario:

• First, resolve the expression \(-(-2) - 5(3)\).
• Begin by noting that \(-(-2)\) simplifies to \(2\) because a double negative turns positive.
• Next, calculate the multiplication \(5 \times 3\) yielding \(15\).
• Finally, conduct the subtraction \(2 - 15\) to reach the result of \(-13\).

Vigilance in applying these operations ensures precise and accurate results. Coupled with substitution, these operations become powerful in breaking down and solving complex expressions.

Algebraic Simplification

Algebraic simplification involves reducing expressions into their simplest or most manageable form. This often includes removing parentheses, combining like terms, and carrying out arithmetic operations. Simplification is crucial in making expressions easier to understand and solve.
In our problem, after substitution, the expression becomes more numerical, allowing simplification to take place:

• Start with \(2 - 5 \times 3\). Performing the multiplication \(5 \times 3\) equates to \(15\), simplifying the expression to \(2 - 15\).
• The final simplification is the subtraction, \(2 - 15\), which results in \(-13\).

Simplification involves these careful steps of executing operations in the right sequence, ultimately making it easier to follow and solve. The process highlights how breaking down complex expressions into basic arithmetic can demystify and yield solutions readily. It's a skill essential for solving most algebraic problems effectively.