Problem 39 - Prealgebra

Question

Suppose \(y=3 x-2 .\) Find \(y\) if: $$x=3$$

Step-by-Step Solution

Answer

If \( x = 3 \), then \( y = 7 \).

Step 1: Identify Given Equation

The equation for this problem is given as \( y = 3x - 2 \). This is a linear equation that expresses \( y \) in terms of \( x \).

Step 2: Substitute Value of x

We're asked to find \( y \) when \( x = 3 \). So, we substitute \( x = 3 \) in the equation, making it \( y = 3(3) - 2 \).

Step 3: Calculate the Product

Calculate the multiplication of \( 3 \times 3 = 9 \). The equation now simplifies to \( y = 9 - 2 \).

Step 4: Perform the Subtraction

Subtract 2 from 9 to find \( y \). The calculation is \( 9 - 2 = 7 \).

Step 5: Conclude the Result

After substitution and calculation, we find that the value of \( y \) is 7 when \( x = 3 \).

Key Concepts

Solving Linear Equations

Linear equations are equations where each term is either a constant or the product of a constant and a single variable. The power of the variable is always one. For example, in the equation \( y = 3x - 2 \), \( y \) and \( x \) are variables, and 3 and -2 are constants. The goal in solving a linear equation is to find the value of one variable when the value of the other is given.

To solve a linear equation like the one in the exercise, follow these steps:

• Identify the given linear equation.
• Determine the value you need to find (either \( x \) or \( y \)).
• Substitute any known values into the equation.
• Simplify the equation to solve for the unknown variable.

Solving linear equations is fundamental in algebra and forms the basis for more complex equations.

Substitution Method

The substitution method is a technique used to solve equations, especially linear ones. Here, you "substitute" one value into an equation to find the other unknown. This is particularly useful when you have one variable defined in terms of another. In our exercise, \( y = 3x - 2 \) defines \( y \) in terms of \( x \).

The steps for applying the substitution method are:

• Identify the known variable. In our case, this is \( x = 3 \).
• Substitute this value into the equation in place of the corresponding variable.
• Perform the arithmetic operations to solve for the unknown variable.

Using substitution simplifies the problem, making it easier to solve linear equations accurately and efficiently.

Prealgebra Problems

Prealgebra is a stage of mathematics that prepares students for algebra. It involves the basics of arithmetic and an introduction to various algebraic concepts. Problems in prealgebra usually have you working with simple equations like the one presented in the exercise, \( y = 3x - 2 \).

When tackling prealgebra problems, remember:

• Understand what each term in an equation represents.
• Know basic operations, such as addition, subtraction, multiplication, and division.
• Familiarize yourself with solving for unknowns using substitution or other methods.

Prealgebra lays the groundwork for all subsequent math studies, making it essential to grasp these foundational concepts early on.