Problem 64 - Algebra 1

Question

Graph the two lines in the same coordinate plane. Then find the coordinates of the point at which the lines cross. \(y=-1, x=0\)

Step-by-Step Solution

Answer

The lines \(y=-1\) and \(x=0\) intersect at the point (0, -1).

Step 1: Understanding the equations

The given equations are \(y=-1\) and \(x=0\). The first equation, \(y=-1\), is a horizontal line that passes through the point (0, -1). The second equation, \(x=0\), is a vertical line that passes through the origin (0, 0).

Step 2: Point of intersection

The point of intersection is where the x-coordinate of the first line equals the x-coordinate of the second line and the y-coordinate of the first line equals the y-coordinate of the second line (where the lines cross). As the point (0, -1) is on the line \(y=-1\) and (0, 0) is on the line \(x=0\), it is clear that these lines intersect at (0, -1). This is because the vertical line \(x=0\) crosses the horizontal line \(y=-1\) at the point where y is -1.

Step 3: Graphing the lines

Both the lines can be drawn on the same coordinate plane to visually confirm the point of intersection. Draw a horizontal line at \(y=-1\) and a vertical line at \(x=0\). The point where these lines cross is the solution, which is (0, -1).

Key Concepts

Coordinate Plane

A coordinate plane is a two-dimensional surface on which points are plotted and located by their positions along two intersecting perpendicular lines. These lines are known as the x-axis and y-axis. The coordinate plane allows us to visually represent mathematical relationships through graphs, such as lines, parabolas, and other shapes. By plotting points and drawing lines on this plane, you can better understand the concept of a graph.

• The horizontal line is called the x-axis, and it runs from left to right.
• The vertical line is the y-axis, and it extends from top to bottom.
• Where these two axes intersect is called the origin.

The origin is the point \(0, 0\), where both the x and y values are zero. Each point on the plane is defined by an ordered pair \(x, y\), where \(x\) is the x-coordinate, and \(y\) is the y-coordinate. This system makes it straightforward to locate points and understand how lines and curves interact on the plane.

Point of Intersection

The point of intersection is where two lines on a coordinate plane cross or intersect each other. This point represents the solution to the system of equations that describes the lines. To find this point, both lines must be considered on the same coordinate plane.

• If you have two lines, say \((y = -1)\) and \((x = 0)\), the intersection is found where both equations are satisfied simultaneously.
• For the line \(y = -1\), the y-coordinate of every point along this line is -1.
• For the line \(x = 0\), the x-coordinate of every point along this vertical line is 0.

In this case, the lines intersect at the point \((0, -1)\). This is the only point where both conditions from the two lines are true, providing a visual and numerical solution to the problem.

Horizontal Line

A horizontal line on the coordinate plane is perfectly flat and parallel to the x-axis. It has the unique property where all points on the line share the same y-coordinate.

• A line described by the equation \(y = c\) is horizontal because the value of y remains constant for all x-values.
• For example, the equation \(y = -1\) means that the line will intersect the y-axis at -1 and run parallel to the x-axis.

This concept is fundamental for distinguishing how different types of lines appear on the graph. The line does not slope up or down but remains at the same height throughout the plane.

Vertical Line

A vertical line is a straight line that runs up and down the coordinate plane, parallel to the y-axis. It is characterized by having the same x-coordinate for any point along the line.

• An equation of the form \(x = d\) represents a vertical line because the value of x is constant for any y-value.
• In our example, the equation \(x = 0\) creates a vertical line passing through the origin, which is typically aligned with the y-axis.

Understanding vertical lines is crucial because they illustrate a foundational geometric concept—lines that intersect the x-axis at precisely one point, maintaining their x-value along all points on the line.