Problem 12 - Algebra 2

Question

Evaluate each expression if \(w=6, x=0.4, y=\frac{1}{2},\) and \(z=-3\). \(z(x+1)\)

Step-by-Step Solution

Answer

The value of the expression is -4.2.

Step 1: Substitute Values

First, substitute the given values for the variables into the expression. The expression given is \(z(x+1)\), and the known values are \(x=0.4\) and \(z=-3\). Substitute these into the expression: \(-3(0.4+1)\).

Step 2: Simplify Inside the Parentheses

Now, simplify the expression inside the parentheses. Add the numbers: \(0.4 + 1 = 1.4\). The expression now becomes: \(-3 \times 1.4\).

Step 3: Perform Multiplication

Finally, multiply \(-3\) by \(1.4\) to evaluate the expression. This calculation gives \(-4.2\).

Key Concepts

Substitution

Substitution is the first step when evaluating any expression. It simply means replacing the variables in an expression with the values assigned to them. This makes the expression numerical, allowing for straightforward arithmetic operations.
For example, if you have an expression like \( z(x+1) \), you start by substituting each variable present. If \( x = 0.4 \) and \( z = -3 \), you replace "\( x \)" and "\( z \)" in the expression with these numbers. Thus, \( z(x+1) \) becomes \(-3(0.4 + 1)\).

• Always double-check you've replaced each variable correctly.
• After substitution, the expression is ready for more actions like simplification or direct calculations.

Simplifying Expressions

Simplifying expressions refers to combining or reducing parts of an expression into a simpler form. It's similar to cleaning up an equation to make solving easier.
When you have an expression like \(-3(0.4 + 1)\), you first look inside the parentheses. Here, you need to add \(0.4\) and \(1\). So, \(0.4 + 1 = 1.4\).

• Always perform operations inside parentheses first, as per the order of operations (PEMDAS/BODMAS).
• Keep your expressions organized by writing each step clearly, which helps avoid errors.

Now, the expression is simplified to \(-3 \times 1.4\), making it easier to solve.

Multiplying Decimals

Multiplying decimals is a key skill when working with numerical expressions. Here, it involves multiplying a negative number by a decimal. Once your expression is simplified to \(-3 \times 1.4\), you're ready to multiply.

• Remember that multiplying two numbers where one is negative gives a negative result.
• When multiplying decimals, treat the numbers like whole numbers first. Multiply \(3\) by \(14\) to get \(42\).
• Then, account for the decimal by placing it correctly in the result. Since there's one decimal place in \(1.4\), the result becomes \(-4.2\).

With practice, multiplying decimals becomes as easy as working with whole numbers. Always ensure to adjust the decimal placement in your final answer.