Problem 27 - Algebra 1: Concepts and Skills

Question

Evaluate the function for the given value of x. $$ y=\sqrt{3 x-5} ; 7 $$

Step-by-Step Solution

Answer

The value for \(y\) when \(x = 7\) is 4.

Step 1: Substitute the Value

Begin by substituting \(x = 7\) into the equation. This gives: \(y=\sqrt{3*7-5}\).

Step 2: Simplify the Expression

Next, simplify the expression inside the square root. This gives: \(y=\sqrt{21-5}\).

Step 3: Calculate the Result

Further simplify the expression. This results into: \(y=\sqrt{16}\).

Step 4: Find the Square Root

Finally, find the square root to get the value for \(y\), which gives: \(y=4\).

Key Concepts

Substitution

Substitution is a key step in math problems involving functions or equations. It's the process of replacing a variable with a given number or expression. In our exercise, we have the function \( y=\sqrt{3x-5} \) and we need to evaluate it for \( x = 7 \).
To do this, you take the value of \( x \) that is provided (which is 7) and substitute it directly into the equation wherever \( x \) appears.
Here's how the substitution step looked in our problem:

• Start with the original equation: \( y=\sqrt{3x-5} \)
• Replace \( x \) with 7: \( y=\sqrt{3(7)-5} \)

This substitution simplifies the evaluation of the function and sets the stage for the next step of simplifying the expression.

Simplifying Expressions

Simplifying expressions is about reducing them into their simplest form. This often involves performing arithmetic operations or combining like terms in an equation.
In the given exercise, after substituting \( x = 7 \) into the equation, we have the expression \( \sqrt{3 \times 7 - 5} \).
Now let's simplify this:

• First, calculate \( 3 \times 7 = 21 \)
• Then solve the subtraction: \( 21 - 5 = 16 \)
• Now the expression is reduced to \( \sqrt{16} \)

Each simplification step makes the problem easier to solve, paving the way to compute the square root."

Square Roots

When you encounter square roots in math, you're dealing with a value that, when multiplied by itself, gives the original number under the square root sign.
In our example, we reached \( \sqrt{16} \) through substitution and simplification.
The task now is to find the square root of 16.

• The square root of 16 is 4 because \( 4 \times 4 = 16 \)

Calculating square roots is a fundamental skill in math that simplifies the evaluation of expressions. Ensuring you're comfortable with basic square roots will make solving such problems much more manageable.

Algebraic Expressions

Algebraic expressions involve variables, coefficients, and operations (like addition and multiplication). They're central to algebra and help represent quantities symbolically.
The function \( y=\sqrt{3x-5} \) in this exercise is an algebraic expression that includes a square root sign and a linear expression inside it.
Algebraic expressions can often be manipulated by using methods like substitution and simplification to find desired values.

• Identify and understand each component: variables, coefficients, and constants.
• Apply operations correctly to simplify the expression.
• Use substitution to evaluate the expression at specific values.

Being proficient with algebraic expressions is crucial because they form the foundation for more complex algebraic manipulations.