Problem 43 - Prealgebra

Question

Suppose \(y=3 x-2 .\) Find \(y\) if: $$x=0$$

Step-by-Step Solution

Answer

When \( x = 0 \), \( y = -2 \).

Step 1: Understand the Given Equation

The equation provided is a linear equation representing a straight line: \( y = 3x - 2 \). This tells us that \( y \) is dependent on the value of \( x \), with a slope of 3 and a y-intercept of -2.

Step 2: Substitute the Given Value of x

We need to determine the value of \( y \) when \( x = 0 \). Substitute \( x = 0 \) into the equation \( y = 3x - 2 \).

Step 3: Perform the Calculation

Substitute the value of \( x \) into the equation: \[ y = 3(0) - 2 \]. Upon simplification, \( 3 \times 0 = 0 \), so the equation becomes \( y = 0 - 2 \).

Step 4: Simplify the Result

Simplify the final expression from the previous step to find: \( y = -2 \).

Key Concepts

Slope in Linear Equations

The slope of a line in a linear equation describes its steepness and direction. For the line given by the equation \( y = 3x - 2 \), the slope is represented by the coefficient of \( x \), which is 3. This tells us that for every 1 unit increase in \( x \), \( y \) increases by 3 units.

In a linear equation, the slope is crucial as it provides insight into how variables are related. A positive slope, like in this case, indicates an upward trend from left to right. Conversely, if the slope were negative, the line would slope downwards.

Understanding slope helps you predict how changes to the independent variable (\( x \)) affect the dependent variable (\( y \)). This is particularly useful in real-world scenarios where relationships between variables need to be analyzed.

The Y-Intercept in Linear Equations

The y-intercept of a line is the point where the line crosses the y-axis. In the equation \( y = 3x - 2 \), the y-intercept is -2. This means when \( x = 0 \), the value of \( y \) is -2.

The y-intercept is a key component of linear equations as it represents the starting point of the line on a graph. It helps set the context for where the line will pass through the y-axis, showing the initial value of \( y \) when no change in \( x \) is considered.

Knowing how to find and interpret the y-intercept is fundamental because it gives a clear starting point from which you can understand the entire linear relationship in the equation.

Understanding the Dependent Variable in Equations

In the equation \( y = 3x - 2 \), \( y \) is considered the dependent variable. The term "dependent variable" means that the value of \( y \) is determined by the value of another variable, in this case, \( x \).

Understanding the dependent variable is essential because it shows the outcome or the result you're predicting or calculating. It is usually found on the vertical axis in graphs, responding to changes in the independent variable along the horizontal axis.

This concept helps in understanding how output depends on input, and in analyzing changes. It aids in predictive modeling, allowing you to calculate how changes to inputs (like \( x \)) change the results (like \( y \)). Being able to identify and manipulate the dependent variable is critical in fields that rely on data analysis and predictive algorithms.