Problem 64 - Prealgebra

Question

Find the value of each of the following expressions when \(x=3 .\) You may substitute 3 for \(x\) in each expression the way it is written, or you may simplify each expression first and then substitute 3 for \(x .\) $$5 x+3 x+2+4$$

Step-by-Step Solution

Answer

The value of the expression when \(x=3\) is 30.

Step 1: Understand the Expression

The expression given is \(5x + 3x + 2 + 4\). We need to find its value when \(x=3\).

Step 2: Combine Like Terms

First, simplify the expression by combining like terms. The terms involving \(x\) are \(5x\) and \(3x\). Combine these to get \((5+3)x = 8x\). Thus, the expression becomes \(8x + 2 + 4\).

Step 3: Simplify Constant Terms

Continue simplifying the expression \(8x + 2 + 4\). Combine the constants: \(2 + 4 = 6\). Now the expression is \(8x + 6\).

Step 4: Substitute Value of x

Substitute \(x=3\) into the simplified expression \(8x + 6\). This gives \(8(3) + 6\).

Step 5: Calculate Result

Calculate the multiplication first: \(8 \times 3 = 24\). Then add the result to the constant: \(24 + 6 = 30\).

Key Concepts

Like Terms

In algebra, "like terms" are terms that contain the same variable raised to the same power. This similarity allows us to combine them easily. For example, in the expression \(5x + 3x + 2 + 4\), the terms involving the variable \(x\) are \(5x\) and \(3x\). Since they both have the variable \(x\) to the first power, they are like terms and can be combined.

Combining like terms simplifies the expression: you simply add or subtract the coefficients (the numerical part of the terms).

• \(5x + 3x = (5+3)x = 8x\)

This process reduces the expression to have fewer terms and makes further calculations easier. Recognizing and combining like terms is a fundamental skill in algebra, as it helps keep expressions neat and efficient.

Substitute Value

Substituting a value into an expression means replacing the variable with a given number. This process helps in evaluating expressions for specific values.

For instance, in our exercise, we were asked to find the value of \(8x + 6\) when \(x = 3\). To do this, you replace every instance of \(x\) in the expression with 3.

• Substituting into the expression gives us \(8(3) + 6\).

After substitute, the expression turns entirely into numbers, which can then be simplified through arithmetic calculations. The ability to substitute a value accurately is crucial as it applies in various mathematical problems and real-world situations.

Simplifying Expressions

Simplifying an expression involves reducing it to its most compact form while still expressing the same condition or value. This process includes combining like terms and performing arithmetic operations.

In our example, we started with the expression \(5x + 3x + 2 + 4\). The process involved:

• Combining like terms: \(5x + 3x = 8x\)
• Combining constants: \(2 + 4 = 6\)

This simplified the expression to \(8x + 6\).

Simplifying makes it easier to handle expressions, especially when further calculations, like substitution, are needed. It allows complex mathematical statements to be more easily interpreted and solved. Mastery of simplifying expressions is also beneficial in problem-solving across various areas in mathematics.