Problem 106 - Introductory Algebra for College Students

Question

I'm working with a linear equation in two variables and found that \((-2,2),(0,0),\) and \((2,2)\) are solutions. When a real-world situation is modeled with a linear cquation in two variables, I can use its graph to predict specific information about the situation.

Step-by-Step Solution

Answer

The linear equation that captures the solutions is \(y = -x\).

Step 1: Establishment of the Formula

Firstly, remember the standard form of a linear equation, \(y = mx + c\), where \(m\) is the slope and \(c\) is the y-intercept.

Step 2: Determination of the Slope

By using any two of the given points, you can figure out the slope. Let's consider \((x1, y1) = (-2, 2)\) and \((x2, y2) = (0, 0)\). The slope \(m\) is given by \((y2 - y1) / (x2 - x1)\). Substituting the points, the slope is \((0 - 2) / (0 - (-2)) = -1\).

Step 3: Finding the y-intercept

You can determine the y-intercept by substituting the slope and one of the points into the equation. Let's substitute \(m = -1\) and the point \((0,0)\) into \(y = mx + c\). Hence, \(c = 0\).

Step 4: Formation of the Linear Equation

With the slope and y-intercept, you can form the equation of the line. Hence, the equation of line is \(y = -x + 0\) or simply \(y = -x\).

Key Concepts

Standard Form of a Linear Equation

The standard form of a linear equation, often taught as a foundational concept in algebra, is an organized way to present equations of straight lines. It's written as \( Ax + By = C \), where \( A \) and \( B \) cannot both be zero. This layout is especially useful for analyzing the equation to predict relationships between variables when faced with real-world scenarios such as budgeting or construction plans.

Essentially, the standard form allows us to easily determine where the line will cross the Y-axis and the X-axis, known as the intercepts. For instance, if \( B \eq 0 \) and \( A = 0 \) the line crosses the Y-axis (y-intercept) and is horizontal; if \( A \eq 0 \) and \( B = 0 \), the line crosses the X-axis (x-intercept) and is vertical. When neither \( A \) or \( B \) equals zero, the line is neither vertical nor horizontal and will intercept both axes at certain points determined by solving \( Ax + By = C \) for \( x \) and \( y \) independently.

In the context of our exercise, the equation \( y = -x \) can be expressed in standard form as \( x + y = 0 \), by adding \( x \) to both sides of the equation. This format underscores the reciprocal relationship between \( x \) and \( y \) values in determining points along the line.

Slope Calculation

The slope of a line is a measure of how steep the line is, which corresponds to the rate of change between the variables \( x \) and \( y \) on a graph. To calculate the slope, denoted by \( m \), you use the coordinates of two distinct points on the line. Formally, the slope formula is \( m = \frac{y2 - y1}{x2 - x1} \), which is the rise over the run between two points \( (x1, y1) \) and \( (x2, y2) \).

A positive slope means that as \( x \) increases, \( y \) also increases, which indicates an upward slant from left to right. Conversely, a negative slope suggests that as \( x \) increases, \( y \) decreases, resulting in a downward slant. A zero slope means the line is horizontal, and an undefined slope (when the denominator is zero due to same \( x \) values) corresponds to a vertical line.

For the given exercise, we derived the slope as \( -1 \) which indicates a consistent decrease in \( y \) with each increase in \( x \) by one unit. This negative slope impacts how the line is graphed and interpreted in the context of any situation it models.

Graphing Linear Equations

Graphing linear equations is an illustrative method to visualize the relationship between variables in a linear equation. It involves plotting points on the coordinate plane and then connecting them to form a straight line. This line is the graph of the linear equation, and each point on the line represents a solution to the equation.

To graph a linear equation, one typically starts by finding the y-intercept (the point where the line crosses the y-axis) and plotting this point. Then, using the slope, which tells you how to move from one point on the line to another, additional points are plotted. The 'rise' over 'run' aspect of the slope dictates the directionality and steepness of the line on the graph.

Step-by-Step Graphing

Starting with the y-intercept, if the slope is negative, you move down (if the rise is negative) or up (if the rise is positive) and then to the right (positive run) by specific units. For example, a slope of \( -1 \) requires moving down 1 unit for every 1 unit moved to the right. This helps in plotting the next point and, consequently, drawing the line. In the exercise's context, starting at point \( (0, 0) \) which is our y-intercept, and applying the slope of \( -1 \) creates a line descending diagonally and thereby graphically displaying the decline represented by the linear equation.