Question

Find the zero of \(f(x)=3 x+7\)

Short answer

The zero of the function is \( x = -\frac{7}{3} \).

Step-by-step solution

Understand the Definition of Zero

The zero of a function is the point where the function value is equal to zero. In other words, solving for the zero of the function means setting the function equal to zero and solving for the variable.

Set the Function Equal to Zero

Start by setting the function equal to zero: \[ f(x) = 3x + 7 \]Set it equal to zero: \[ 3x + 7 = 0 \]

Isolate the Variable x

Subtract 7 from both sides of the equation to isolate the term with x: \[ 3x + 7 - 7 = 0 - 7 \]which simplifies to: \[ 3x = -7 \]

Solve for x

Divide both sides by 3 to solve for x: \[ x = \frac{-7}{3} \]

Key concepts

solving linear equations

Linear equations are equations of the first degree, meaning they have no exponents greater than one. They often appear in the form of ax + b = 0, where aeq0. Solving these equations is a basic and fundamental algebraic skill.
In the given exercise, we have the linear equation 3x + 7 = 0, derived from the function 3x + 7.
To solve it, follow these steps:

• The first step is to recognize that you need to find the value of x that makes the equation true.
• Then, we apply algebraic techniques to isolate the variable x.
• Here, algebraic manipulation helps us to solve for x.

Let’s examine these techniques in subsequent sections.

isolate variable

Isolating the variable is a crucial strategy. It means getting the variable, in this case, x, alone on one side of the equation. For 3x + 7 = 0:

• First, subtract 7 from both sides to remove the constant term:

3x + 7 - 7 = 0 - 7, simplifying to 3x = -7.
• Next, divide both sides by 3 to isolate x:

3x / 3 = -7 / 3, which simplifies to x = -7 / 3.

Isolating the variable helps to simplify complex equations and makes solving them more manageable. Knowing these steps helps in tackling a wide range of algebra problems.

algebraic manipulation

Algebraic manipulation involves using mathematical operations to rearrange equations. In our exercise, we used two fundamental operations:

• Subtraction: To eliminate a constant from one side of the equation.
• Division: To isolate the variable by reducing the coefficient to 1.

For 3x + 7 = 0:
Subtracting 7 from both sides:
3x + 7 - 7 = 0 - 7,
simplifies to 3x = -7.

Dividing both sides by 3:
3x / 3 = -7 / 3, simplifies to x = -7/3.

These operations demonstrate how algebraic manipulation makes it easier to solve for unknowns and find function zeros.

function zeros

Function zeros are the points where the function’s value is zero. Finding these zeros is important in algebra and calculus for analyzing function behavior.
In our given function 3x + 7, the zero is found by solving the equation:
3x + 7 = 0.

• Step 1: Isolate the variable by algebraic manipulation.
• Step 2: Solve the equation to find x = -7/3.

So, the zero of the function is x = -7/3. This is the point where the function intersects the x-axis. Recognizing zeros helps understand roots and solutions in various contexts.