Problem 57 - Prealgebra

Question

Use the formula \(y=\frac{1}{2} x-3\) to find \(y\) if: $$x=0$$

Step-by-Step Solution

Answer

If \( x = 0 \), then \( y = -3 \).

Step 1: Understand the Given Equation

We are provided with the equation \( y = \frac{1}{2}x - 3 \). Our task is to find the value of \( y \) when \( x = 0 \).

Step 2: Substitute the Given Value of x

Replace \( x \) in the equation with \( 0 \). This will change our equation to \( y = \frac{1}{2}(0) - 3 \).

Step 3: Simplify the Equation for y

Calculate the multiplication first, which is \( \frac{1}{2} \times 0 = 0 \). This simplifies the equation to \( y = 0 - 3 \).

Step 4: Perform the Subtraction

Carry out the subtraction \( 0 - 3 = -3 \). Therefore, the value of \( y \) when \( x = 0 \) is \( -3 \).

Key Concepts

Substitution Method

The substitution method is a helpful technique often used in solving equations, especially linear equations. It involves replacing a variable with a given value to simplify the equation and solve for the unknown. This method becomes even more straightforward when dealing with single-variable linear equations. Here's how it generally works:

• First, identify the variable you need to substitute. In our example, it's the variable \( x \).
• Take the given value for this variable (here, \( x = 0 \)) and substitute it directly into the equation.
• This substitution replaces the variable with a number, making the equation much easier to work with.

In the example provided, the equation was \( y = \frac{1}{2}x - 3 \). By substituting \( x = 0 \), we transformed it into \( y = \frac{1}{2}(0) - 3 \), a much simpler equation to handle.
The substitution method is not only efficient but also systematically leads to solving equations step by step. It's a fundamental concept in algebra, laying the groundwork for more complicated problems.

Simplifying Expressions

Simplifying expressions means breaking down a mathematical equation or expression into its simplest form. This involves performing basic arithmetic operations or using algebraic identities to make it easier to interpret and solve. In the context of our linear equation, simplifying the expression after substitution is crucial:

• Once you've substituted the values, carry out any multiplications or divisions first in the expression.
• Next, perform any additions or subtractions to further simplify the expression.

For instance, after substituting \( x = 0 \) into the equation \( y = \frac{1}{2}x - 3 \), you simplify the expression by performing the multiplication \( \frac{1}{2} \times 0 = 0 \).
This step drastically simplifies the equation to \( y = 0 - 3 \). From here, straightforward subtraction yields the final answer. Simplifying expressions is a vital skill in algebra, ensuring that equations are easily manageable and understandable.

Algebraic Manipulation

Algebraic manipulation involves rearranging or reworking expressions and equations using a variety of algebraic techniques. This is often necessary when trying to isolate a variable, simplify an equation, or solve for unknowns. In our exercise, algebraic manipulation is applied after substituting the given \( x \) value:

• Initially, we substitute \( x = 0 \) into \( y = \frac{1}{2}x - 3 \), resulting in \( y = 0 - 3 \).
• Next, apply the rules of arithmetic by performing subtraction: \( 0 - 3 \).

Through these straightforward manipulations, you arrive at the solution \( y = -3 \).
Algebraic manipulation allows for the effective solution of algebraic equations, enabling students to transition seamlessly from problem setup to solving. By understanding how to manipulate equations, students can approach most algebra problems with confidence.