Problem 15 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the \(x\) - and \(y\) -intercepts of the graph of the equation, if possible. $$x y-2 y-x+1=0$$

Step-by-Step Solution

Answer

The x-intercept of the graph is at \(x = 1\) and the y-intercept is at \(y = 0.5\).

Step 1: Solve for the x-intercept

First, set \(y=0\) in the given equation \(xy - 2y - x + 1 = 0\) to find the x-intercept. The equation becomes \(0 - 0 - x + 1 = 0\), simplify this to find \(x = 1\). Therefore, the x-intercept of the graph is at \(x = 1\).

Step 2: Solve for the y-intercept

Next, set \(x=0\) in the equation to find the y-intercept. The equation becomes \(0 - 2y + 0 + 1 = 0\). Simplify to solve for \(y\), so here \(y = 0.5\). Therefore, the y-intercept of the graph is at \(y = 0.5\).

Key Concepts

X-Intercept

The x-intercept is a point on the graph where the line crosses the x-axis. To find it, we set the y value to zero in the equation and solve for x.

By substituting y=0, we neutralize the impact of y in the equation, hence the solution purely reflects the value of x when the line touches the x-axis. In the given exercise, setting y to zero simplified the equation to -x + 1 = 0, indicating the x-intercept is at the point (1, 0).

Understanding x-intercepts is essential in graphing linear equations and analyzing the point where the function has zero vertical height from the x-axis.

Y-Intercept

Conversely, the y-intercept is found by setting x to zero in the equation and solving for y. This gives us the point where the line intersects with the y-axis.

When we plug in x=0, the terms involving x in our equation are eliminated, leaving us with a straightforward equation to find the value of y. In the exercise, substituting x with zero lead to the equation -2y + 1 = 0 and solving this resulted in y = 0.5, or the y-intercept at (0, 0.5).

Identifying y-intercepts is crucial for graphing and understanding the initial value of the function when the literal horizontal shift is absent.

Equation of a Line

The equation of a line is a formula that demonstrates the relationship between the x and y coordinates of all the points on a line. One of the most common forms of the equation is the slope-intercept form, written as y = mx + b, where m represents the slope of the line and b stands for the y-intercept.

Our exercise features a more complex equation which still represents a line. To determine the intercepts, we manipulate the equation to solve for one variable at a time. Grasping the concept of a line's equation allows us to predict the line's behavior, analyze its steepness, and determine its intercepts with the axes.

Manipulating an equation to standard form, slope-intercept form, or point-slope form can greatly facilitate understanding and graphing linear relationships.

Graphing Linear Equations

Graphing linear equations is a means of visualizing the relationship expressed by a linear equation. You plot the line on a coordinate plane using important features like intercepts and slope.

To graph a linear equation, find the x and y intercepts as illustrated in prior steps. Plot these intercepts on the graph, then draw a straight line through the points to represent the equation's all possible solutions.

A well-drawn graph provides a quick visual insight into the characteristics of the line, including the direction it moves, where it increases or decreases, and its intercepts. Graphing is not just a key skill in algebra but also a powerful tool for interpreting and predicting relationships in various fields such as economics, engineering, and the physical sciences.