Problem 9 - Elementary Algebra

Question

Observe the equations and state the relationship being expressed. $$ y=x-7 $$

Step-by-Step Solution

Answer

Answer: The relationship between x and y in the given equation is linear, with a slope of 1 and a y-intercept of -7. This means that for every one unit increase in x, the value of y increases by one unit, and the line passes through the point (0, -7) on the y-axis.

Step 1: Identify the form of the equation

The equation is given as $$y = x - 7$$, which is a linear equation in two variables, x and y. The relationship between x and y can be determined by analyzing the equation.

Step 2: Examine the coefficients and the constants

In the equation $$y = x - 7$$, the coefficient of x is 1, and the constant is -7. The coefficient of x tells us the slope of the line, and the constant term will help us find the y-intercept.

Step 3: Identify the slope

In this equation, the coefficient of x is 1, which represents the slope (m) of the line. This means that for every one unit increase in x, the value of y also increases by one unit.

Step 4: Identify the y-intercept

The constant term in the equation is -7, which represents the y-intercept (b) of the line. This means that when x = 0, the value of y is -7. So the point (0, -7) lies on the line.

Step 5: State the relationship

The given equation represents a linear relationship between x and y, with a slope of 1 (meaning the line is increasing at a constant rate) and a y-intercept of -7 (meaning the line crosses the y-axis at point (0, -7)). The equation $$y = x - 7$$ describes this relationship.

Key Concepts

Understanding the Slope in Linear Equations

When we talk about the slope in a linear equation like \( y = x - 7 \), we refer to how steep the line is. The slope indicates how much the value of \( y \) changes when \( x \) increases by one unit. In the equation \( y = x - 7 \), the slope is 1 because of the coefficient of \( x \). This means:

• For every 1 unit increase in \( x \), \( y \) increases by 1 unit as well.
• The relationship between \( x \) and \( y \) is direct and proportionate.

A slope of 1 indicates a 45-degree angle line, showing a perfect balance between \( x \) and \( y \). If the slope were greater than 1, the line would be steeper, and if less than 1, it would be flatter. It's crucial to identify and understand the slope to predict how changes in \( x \) influence \( y \).

Exploring the Y-Intercept

The y-intercept is another vital component of a linear equation like \( y = x - 7 \). It tells us where the line crosses the y-axis. In this equation, the y-intercept is -7, meaning:

• When \( x = 0 \), \( y = -7 \).
• The line crosses the y-axis at point \((0, -7)\).

Understanding the y-intercept helps us identify the starting point of the line on a graph. It's the value of \( y \) when \( x \) is zero, giving us a concrete point to anchor our line. This is especially useful when plotting the line on a coordinate system, as you can start at the y-intercept and use the slope to draw the line accurately.

Linear Equations with Two Variables

Linear equations frequently involve two variables, typically \( x \) and \( y \). These equations form straight lines when graphed. In the equation \( y = x - 7 \):

• \( x \) is the independent variable, meaning it can vary freely.
• \( y \) is the dependent variable, calculated based on \( x \).

This equation shows a direct linear relationship between \( x \) and \( y \). By assigning different values to \( x \), we can determine corresponding values of \( y \). Both variables work together, with the slope dictating how one changes in relation to the other. Linear equations in two variables are fundamental in algebra, utilized in many real-world applications like modeling economic trends or predicting scientific outcomes.