Problem 67 - 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 \(x .\) $$3 x-8+2 x-3$$

Step-by-Step Solution

Answer

The value of the expression when \(x = 3\) is 4.

Step 1: Identify Terms

The given expression is \(3x - 8 + 2x - 3\). First, identify all the terms that include the variable \(x\) and the constant terms.

Step 2: Simplify the Expression

Combine like terms in the expression. The terms with \(x\) are \(3x\) and \(2x\). Add them together: \(3x + 2x = 5x\). The constant terms are \(-8\) and \(-3\), which combine to \(-11\). So the expression simplifies to \(5x - 11\).

Step 3: Substitute and Calculate

Substitute \(x = 3\) into the simplified expression \(5x - 11\). This gives us \(5(3) - 11\). First, calculate \(5 \times 3 = 15\). Then, subtract 11: \(15 - 11 = 4\).

Key Concepts

Substitution Method

The substitution method is a valuable technique used to evaluate expressions. Here, you replace variables within an expression with given numerical values. In the problem, you're asked to substitute the value of 3 for the variable \(x\) in the expression \(3x - 8 + 2x - 3\). This conversion allows you to work with numbers exclusively. This approach simplifies complex expressions by turning them into basic arithmetic problems.
- Start by taking the original expression, as given, and directly replace \(x\) with 3 throughout. - This means our expression becomes \(3(3) - 8 + 2(3) - 3\).- Perform calculations step-by-step: - Compute \(3 \times 3 = 9\) and \(2 \times 3 = 6\). - Then substitute these values back, simplifying to \(9 - 8 + 6 - 3\).- Finally, solve this straightforward arithmetic step to find the expression's value.Using the substitution method not only saves time but also makes understanding complex expressions easier for students. It transforms the problem into something more manageable.

Combining Like Terms

Combining like terms is the act of gathering similar elements in an expression. This makes it simpler and reduces its complexity. In our example, the expression \(3x - 8 + 2x - 3\) includes several terms that we can combine. Here's how it's done:
- Locate terms that have the same variable. For \(3x - 8 + 2x - 3\), note the terms with \(x\): \(3x\) and \(2x\).- Combine these by adding their coefficients: \(3 + 2 = 5\), yielding \(5x\).- Next, handle the constant terms, which here are \(-8\) and \(-3\).- Adding these constants gives \(-8 + (-3) = -11\).- The expression simplifies to \(5x - 11\).By focusing on combining like terms, you reduce complex expressions into fewer terms, making substitution and further calculations much more straightforward.

Variable Expressions

Variable expressions include one or more variables along with potential coefficients and constants. They require an understanding of variable roles in algebraic expressions. For the expression \(3x - 8 + 2x - 3\), we're dealing with both variable and constant terms. Understanding this distinction is crucial:
- **Variable Terms:** These depend on the value of \(x\). Here, \(3x\) and \(2x\) are terms, meaning they change depending on what \(x\) equals.- **Constant Terms:** Numbers that do not change when substituting values for the variable. In this case, \(-8\) and \(-3\) remain the same.Finding the value of such expressions involves determining the effect variables have when substituted by a number, alongside the constant contribution. Being comfortable with how variables interact in expressions strengthens your mathematical skills and ensures you're ready to tackle more advanced algebraic concepts.