Problem 7 - Fundamentals of Mathematics

Question

Find the value of each expression. $$-3 x-5 y+2 z, \text { if } x=-4, y=3, z=0$$

Step-by-Step Solution

Answer

The value of the expression is -3.

Step 1: Substitute Values

Start by substituting the given values for the variables into the expression. The expression is \(-3x - 5y + 2z\), and the values you have are \(x = -4\), \(y = 3\), and \(z = 0\). Replace these in the expression: \(-3(-4) - 5(3) + 2(0)\).

Step 2: Simplify Each Term

Simplify each term in the expression: 1. Calculate \(-3(-4)\), which equals \(12\). 2. Calculate \(-5(3)\), which equals \(-15\). 3. Calculate \(2(0)\), which equals \(0\).

Step 3: Sum the Results

Add the results from each term's simplification: \(12 + (-15) + 0\).Then calculate step-by-step: \(12 - 15 = -3\), and \(-3 + 0 = -3\). The entire expression simplifies to \(-3\).

Key Concepts

Substitute Values

When you're working with algebraic expressions, one of the essential skills you need is substituting values for variables. This involves taking the given values for each variable and inserting them into the expression wherever that variable appears. For example, if you're given the expression \(-3x - 5y + 2z\), and the values \(x = -4\), \(y = 3\), and \(z = 0\), you'll replace each variable with its corresponding value.

• For \(x\): replace \(x\) with \(-4\).
• For \(y\): replace \(y\) with \(3\).
• For \(z\): replace \(z\) with \(0\).

This process transforms the expression into \(-3(-4) - 5(3) + 2(0)\). Substituting values correctly is crucial as it sets the foundation for simplifying terms later.

Simplify Terms

After substituting the values, the next step is simplifying each term in the expression. This step involves performing arithmetic operations to reduce each part of the expression to its simplest form. For the expression \(-3(-4) - 5(3) + 2(0)\):

• For the first term, \(-3(-4)\): Multiply \(-3\) by \(-4\) to get \(12\). Multiplying two negative numbers results in a positive product.
• For the second term, \(-5(3)\): Multiply \(-5\) by \(3\), resulting in \(-15\). A negative multiplied by a positive number results in a negative product.
• For the third term, \(2(0)\): Multiply any number by \(0\) and the result is \(0\).

Reaching this simplified version of all terms is essential for accurate evaluation of the overall expression.

Evaluate Expressions

Once each term has been simplified, the final step is to evaluate the entire algebraic expression by combining these simplified results. Using our example, we simplified to the terms \(12\), \(-15\), and \(0\). Now, we need to add these values together:

• First, combine \(12\) and \(-15\). Adding \(-15\) to \(12\) gives \(-3\). It's essentially subtracting \(15\) from \(12\) due to the negative sign.
• Next, add \(0\) to \(-3\), which still results in \(-3\).

The evaluated expression is \(-3\). This step compiles all your work into a single, clear result, completing the evaluation process.