Problem 40 - Prealgebra

Question

Suppose \(y=3 x-2 .\) Find \(y\) if: $$x=-5$$

Step-by-Step Solution

Answer

If \(x = -5\), then \(y = -17\).

Step 1: Understand the given equation

The equation given in the problem is a linear equation: \( y = 3x - 2 \). This equation shows that for any input value of \( x \), the output \( y \) is calculated by multiplying \( x \) by 3 and then subtracting 2 from the result.

Step 2: Substitute the given value of 'x' into the equation

Since the problem states that \( x = -5 \), we need to substitute \( -5 \) into the equation in place of \( x \). The equation then becomes:\[ y = 3(-5) - 2 \]

Step 3: Perform the multiplication

Multiply \( 3 \) by \( -5 \) to solve the first part of the equation. This gives:\[ 3 \times (-5) = -15 \]

Step 4: Perform the subtraction

Now, take the result from the multiplication \( -15 \) and subtract 2 from it:\[ y = -15 - 2 \]

Step 5: Simplify to find 'y'

Solve the subtraction to find the value of \( y \):\[ y = -17 \]

Key Concepts

Substitution Method

The substitution method is a technique used to solve equations, especially useful in solving systems of equations or when handling linear equations like the one in the exercise:
- You are given an equation, in this case, \( y = 3x - 2 \), with a particular value for \( x \). These types of equations often exist where the value of one variable depends on another.
- The key is to replace the given value of \( x \) directly into the equation. In this problem, you substitute \( -5 \) for \( x \) into the equation, transforming it to \( y = 3(-5) - 2 \).
- By substituting, you effectively reduce the problem from one of variables to simple arithmetic calculations. This makes it much easier to solve, as you turn the equation into a sequence of simple arithmetic steps.

In general, the substitution method is a powerful tool because:

• It simplifies complex equations into basic arithmetic processes.
• It is versatile and can be applied to many mathematical scenarios.
• It demonstrates the relationship between variables clearly, which can aid in understanding and analyzing mathematical models.

Solving Linear Equations

Solving linear equations involves finding the value of unknown variables that make the equation true. Linear equations are equations of the first degree, meaning they only involve the first power of the variable. The linear equation presented in the exercise is:
- \( y = 3x - 2 \), which represents a straight line when plotted on a graph.

To solve:

• First, understand what each part of the equation represents. In \( y = 3x - 2 \), \( 3x \) shows the relationship between \( x \) and \( y \), stating that \( y \) changes with changes in \( x \).
• The \(-2\) is the y-intercept, which determines where the line crosses the y-axis.
  - Solving this particular linear equation requires substituting a given value for \( x \) to find \( y \).
• This involves performing basic arithmetic operations following the substitution to reach the solution.

Linear equations are fundamental in algebra because:

• They model real-world situations.
• They are the basis of more complex algebraic concepts.
• Mastery of solving them is crucial for progressing in math, science, and engineering fields.

Arithmetic Operations

Arithmetic operations are basic mathematical processes that include addition, subtraction, multiplication, and division. They are crucial when dealing with linear equations as they form the backbone of solving these equations. When you substituted \( x = -5 \) into the equation \( y = 3x - 2 \):

• The first arithmetic operation applied was multiplication: \( 3 \times (-5) \), which gives \(-15 \).
• Next, was subtraction, where you take \(-15\) and subtract \(2\), resulting in \(-17\).

These operations are essential as they allow you to handle and simplify equations to arrive at the correct solution. Key points include:

• Multiplication can change the sign of a number. For example, a positive multiplied by a negative gives a negative result.
• Subtraction is used to adjust the value of a number by decreasing it. If starting with a negative number, subtracting further decreases the value.

Proficiency in arithmetic operations is fundamental:

• It applies across all areas of mathematics.
• It is vital for precision in calculations.
• Understanding arithmetic helps solve and simplify equations effectively.

Grasping these basic operations makes solving equations more intuitive and error-free.