Problem 54 - Prealgebra

Question

Find the value of each expression when \(x=-4\) $$5 x+8$$

Step-by-Step Solution

Answer

The value is -12.

Step 1: Substitute the given value

We start by substituting the given value of \( x = -4 \) into the expression. The original expression is \( 5x + 8 \). After substitution, the expression becomes \( 5(-4) + 8 \).

Step 2: Multiply the coefficient by the variable

Now, we multiply \( 5 \) by \( -4 \). This results in \( 5 \times -4 = -20 \). The expression now is \( -20 + 8 \).

Step 3: Solve the remaining arithmetic

The next step is to add \( -20 \) and \( 8 \). This gives us \( -20 + 8 = -12 \). So, the value of the expression when \( x = -4 \) is \( -12 \).

Key Concepts

Substitution in Algebraic Expressions

Substitution is a fundamental concept in algebra that helps simplify expressions by replacing variables with numbers. It's like filling in the blanks with values you know. In our exercise, we were given the expression \(5x + 8\) and asked to evaluate it when \(x = -4\). To do this, we replace every instance of \(x\) in the expression with \(-4\). So, \(5x + 8\) becomes \(5(-4) + 8\). Substitution is particularly useful because it allows us to transform an expression with unknowns into a more manageable numerical expression, which can then be solved using arithmetic operations. Remember:

• Replace the variable with the given value.
• Ensure you correctly maintain the signs, especially with negative values.
• Write down each step to avoid errors.

In algebra, mastering substitution sets the stage for understanding more complex processes, like solving equations.

Arithmetic Operations with Integers

Once substitution is completed, the next step involves arithmetic operations—essentially, basic math calculations like addition, subtraction, multiplication, and division. In our given problem, after substitution, the expression turned into \(5(-4) + 8\).First, we multiply the integer \(5\) by \(-4\). Since multiplying a positive number by a negative number results in a negative number, we have \(5 \times -4 = -20\). It's essential to be comfortable with the rules for multiplying integers:

• A positive times a positive is positive.
• A negative times a negative is positive.
• A positive times a negative is negative.

Following the multiplication, we move on to addition: \(-20 + 8\). Here, imagine combining positive and negative amounts. Add the positive and negative, treating the operation like moving along a number line. Starting at \(-20\), moving 8 units towards the positive, we land on \(-12\). Arithmetic operations are the backbone of many algebraic processes, so understanding them thoroughly aids in dealing with all levels of math problems.

Prealgebra Concepts

Prealgebra introduces basic mathematical principles and prepares students for more advanced algebra. It generally covers fundamental operations and their properties, understanding variables, and simple equation solving. In the context of our exercise, we focused on finding an expression's value using substitution and arithmetic operations, both vital prealgebra skills. Mastering prealgebra:

• Builds confidence in handling variables.
• Improves numerical problem-solving skills.
• Paves the way for algebraic thinking.
• Enhances understanding of practical situations requiring math.

The goal of prealgebra is to lay a sturdy foundation in these concepts, ensuring that when more complex algebraic ideas are introduced, students have a solid groundwork to build upon. It's about making sure these foundational skills become intuitive and automatic, so you are ready to tackle high school math and beyond.