Problem 11 - Algebra and Trigonometry

Question

11–18 ? Find the x- and y-intercepts of the graph of the equation. $$ y=x-3 $$

Step-by-Step Solution

Answer

x-intercept: (3,0); y-intercept: (0,-3)

Step 1: Find the y-intercept

To find the y-intercept, set \( x = 0 \) in the equation \( y = x - 3 \). Substituting gives \( y = 0 - 3 \), which simplifies to \( y = -3 \). So, the y-intercept is at the point \((0, -3)\).

Step 2: Find the x-intercept

To find the x-intercept, set \( y = 0 \) in the equation \( y = x - 3 \). This gives the equation \( 0 = x - 3 \). Solve for \( x \) by adding 3 to both sides, yielding \( x = 3 \). Thus, the x-intercept is at the point \((3, 0)\).

Key Concepts

Understanding the X-Intercept

The x-intercept of a linear equation is an invaluable concept in mathematics, especially when exploring the relationships depicted on a graph of a line. Essentially, the x-intercept is the point at which the graph of the equation crosses the x-axis.
To find this point, you set the y-value to zero and solve for x. For example, when given the equation of a line, such as \( y = x - 3 \), you would find the x-intercept by substituting 0 for y: \( 0 = x - 3 \).
Through this substitution, you solve for x and discover that the x-intercept is at the point (3, 0). Some important tips to remember about x-intercepts include:

• Set y to zero in the equation to find the x-intercept.
• Solve the resulting equation for x.
• The x-intercept is always represented as a point where y is zero, such as (x, 0).

Recognizing x-intercepts helps in understanding where a graph touches the x-axis, making it a crucial part of graphing linear equations.

Unpacking the Y-Intercept

The y-intercept is another fundamental component when working with linear equations. It indicates the specific point where the line crosses the y-axis. Much like the x-intercept, finding the y-intercept requires a strategic approach of substituting a value in the equation.
For the equation \( y = x - 3 \), to find the y-intercept, you set the x-value to zero. Substituting this into the equation gives \( y = 0 - 3 \), which simplifies to \( y = -3 \). Thus, the y-intercept for this equation is located at the point (0, -3).
Key facts about y-intercepts include:

• Set x to zero in the equation to determine the y-intercept.
• The y-value you solve for is the y-intercept.
• It is represented as a point where x is zero, such as (0, y).

Understanding y-intercepts is essential for interpreting how linear equations behave as they cross vertical axes.

Simplifying Graphing Linear Equations

Graphing linear equations becomes much more accessible once you grasp the concepts of x- and y-intercepts. These intercepts provide foundational points that can be plotted to draw a line representing the equation on a graph.
For example, with the equation \( y = x - 3 \), you can graph this line using the intercepts you’ve found:- The x-intercept (3, 0)- The y-intercept (0, -3)
To draw the line, you plot these intercepts on a coordinate plane and simply draw a line through them, which will accurately represent the equation. Here are steps to efficiently graph linear equations using intercepts:

• Find both the x-intercept and y-intercept of the equation.
• Plot these points on the Cartesian plane.
• Draw a straight line through these points as lines are straight in linear equations.

This visual representation allows for a clearer understanding of the equation's behavior, making graphing an indispensable tool for visualizing algebraic relationships.