Problem 46 - Algebra 1: Concepts and Skills

Question

Evaluate the expression for the given value of the variable. \(2 x^{2}\) when \(x=7\)

Step-by-Step Solution

Answer

The value of the expression \(2x^{2}\) when \(x=7\) is \(98\).

Step 1: Substitute \(x\) with 7

The task is to find the value of the expression \(2x^{2}\) when \(x=7\). So substitute \(x\) in the equation with 7. After this substitution, the equation will look like this: \(2*(7)^{2}\).

Step 2: Square the Value of \(x\)

The equation is now down to \(2*(7)^{2}\). The exponent tells you to multiply \(7\) by itself. So, \(7^{2} = 7*7 = 49\). Now, the equation has simplified to: \(2*49\).

Step 3: Multiply the remained number by 2

The final stage is to multiply \(49\) by \(2\). So, \(2*49 = 98\).

Key Concepts

Substitution Method

The substitution method is a fundamental technique in algebra that simplifies the process of solving expressions or equations. It involves replacing variables with given values to transform an abstract expression into a concrete numerical expression. This allows you to find the specific value or solution to a problem.
To apply the substitution method effectively, follow these simple steps:

• Identify the variable that needs substitution—in this case, the variable is \(x\).
• Replace the identified variable with its specific value. For our exercise, substitute \(x\) with 7 forming the new expression: \(2*(7)^{2}\).
• Once substitution is complete, simplify the expression further, moving on to other mathematical operations like exponentiation or multiplication.

By substituting \(x\) with a numeric value, we make the problem much easier to handle, essentially turning an algebraic expression into a purely numerical one that we can then solve using basic arithmetic.

Exponents

Exponents indicate the number of times a base number is multiplied by itself. In the expression \(x^{2}\), the exponent \(2\) tells us to multiply the base \(x\) twice. When evaluating an expression using exponents, it's crucial to focus on this repeated multiplication aspect.
In our example, after substituting \(x\) with \(7\), the expression becomes \(7^{2}\). This means:

• Calculate \(7 * 7\), as the 2 indicates the base is used twice in the multiplication.
• The result of \(7 * 7\) is \(49\), simplifying the expression further to \(2 * 49\).

Exponents simplify repeated multiplication, making it more manageable. They are a powerful tool for representing large numbers compactly, especially when dealing with variables.

Multiplication in Algebra

Multiplication is a basic arithmetic operation that frequently appears in algebra, often in the context of combining terms or simplifying expressions. In algebraic expressions like \(2x^{2}\), multiplication connects coefficients with variables.

• After substituting \(x\) and calculating the exponent, you are left with \(2 * 49\). Here, multiplication acts as a way to merge the coefficient 2 with the result from the exponent calculation \(49\).
• To solve \(2 * 49\), simply multiply the two numbers: \(2 * 49 = 98\).

In algebra, multiplication is often embedded in more complex expressions and requires precision, especially when combined with other operations like exponents and addition. It is vital to follow the order of operations correctly to avoid any errors and arrive at the correct solution.