Problem 41 - Prealgebra

Question

Suppose \(y=3 x-2 .\) Find \(y\) if: $$x=-\frac{1}{3}$$

Step-by-Step Solution

Answer

The value of \(y\) is \(-3\).

Step 1: Substitute the value for x

We are given the equation \(y = 3x - 2\) and need to find the value of \(y\) when \(x = -\frac{1}{3}\). First, substitute \(-\frac{1}{3}\) into the equation for \(x\). This gives us \(y = 3(-\frac{1}{3}) - 2\).

Step 2: Simplify the equation

Calculate \(3 \times -\frac{1}{3}\). Multiply the outside number by the numerator, which gives us \(-1\). So our equation rewrites to \(y = -1 - 2\).

Step 3: Final calculation

Now, simply solve \(-1 - 2\) to get \(-3\). Thus, \(y = -3\).

Key Concepts

Substitution Method

Substitution is a powerful technique in algebra that helps solve equations by replacing one variable with a known value or another expression. It’s like putting a piece of a puzzle in the right place. In this method, you identify parts of an equation that can be replaced with something known. For the equation provided, \( y = 3x - 2 \), we were given the value of \( x \), specifically \( x = -\frac{1}{3} \).

• Start by substituting the given value into the equation where the variable appears.
• This helps transform the equation into something simpler that is easier to solve for the unknown variable.
• Here, by substituting \( x \) with \( -\frac{1}{3} \), the expression becomes: \( y = 3(-\frac{1}{3}) - 2 \).

This approach is commonly used not only in algebra but also in higher mathematics, where dealing with one unknown at a time makes the problem more manageable.

Algebraic Expressions

Algebraic expressions are combinations of numbers, variables, and operations (like addition, subtraction, and multiplication). They are the building blocks of algebra and are essential in formulating mathematical relationships. In the equation \( y = 3x - 2 \):

• \( 3x \) is an algebraic term where 3 is a coefficient and \( x \) is a variable.
• The term \(-2\) is a constant, simply a number without a variable attached.
• An expression tells you what calculations you must perform on the given values of \( x \) to find \( y \).

It's important to understand how coefficients and constants work because they dictate the relationships between variables. Coefficients describe the relationship's strength or magnitude, and constants adjust the expression’s value.
We simplified \( y = 3(-\frac{1}{3}) - 2 \) to \( y = -1 - 2 \), maintaining the balance of the equation and helping us proceed toward a solution.

Negative Numbers

Negative numbers are less than zero and represented with a minus (-) sign. They’re like the opposite side of a number line from positive numbers and are crucial for mathematical operations. Solving equations with negative numbers requires careful attention:

• When multiplying a positive number by a negative number, the result is negative. In our example, \( 3 \times -\frac{1}{3} \) becomes \(-1\).
• Subtracting a positive number from a negative makes the number even smaller. For instance, \(-1 - 2 = -3\).
• It helps to think of subtraction of negatives as a directional move on a number line; moving left indicates further into negative values.

Students should practice operating with negative numbers to become comfortable with their behaviors in equations. Accurate calculation with negative numbers is key to getting the correct solution in algebraic problems.