Question

Graph the two lines in the same coordinate plane. Then find the coordinates of the point at which the lines cross. \(y=11, x=-8\)

Short answer

The intersection point of the lines \(y=11\) and \(x=-8\) is \(-8, 11\).

Step-by-step solution

Graph the horizontal line

To graph the line \(y=11\), plot a horizontal line that runs through the value 11 on the y-axis.

Graph the vertical line

To graph the line \(x = -8\), plot a vertical line that runs through the value -8 on the x-axis.

Find the intersection

The point where these two lines intersect will be the coordinates that satisfy both equations. Here, that intersection occurs at point \(-8, 11\).

Key concepts

Coordinate Plane

Imagine a large flat surface where each point is defined by a pair of numbers, known as coordinates. That's what the coordinate plane is all about. The plane is divided into four quadrants by two perpendicular lines: the x-axis, which runs horizontally, and the y-axis, which runs vertically. These axes intersect at a point called the origin, denoted as (0,0).

When graphing linear equations or any function, the coordinate plane serves as the stage where the mathematical drama unfolds. The position of a point on this plane is always described by an ordered pair \( (x, y) \) – the x-coordinate first, followed by the y-coordinate. Each coordinate indicates how far along the axis the point is. If x is positive, the point is to the right of the origin; if negative, to the left. Similarly, a positive y-value places the point above the origin, and a negative one below it.

In graphing, using a clear and accurately labeled coordinate plane is crucial. It ensures that functions and points are depicted precisely, assisting in visual learning and understanding of concepts like slope and intercepts.

Intersection of Lines

When two lines share a single point, they intersect. The 'intersection of lines' refers to this exact point shared by both lines. In the case of linear equations, each line represents all the points that satisfy the equation. So, when we graph two or more lines on the same coordinate plane, the intersection is the set of coordinates that satisfy all the equations simultaneously.

Let's take the equations \( y = 11 \) and \( x = -8 \). These are special cases of linear equations, representing a horizontal and a vertical line, respectively. A horizontal line means that the \( y \) value remains constant, whereas for a vertical line the \( x \) value is constant, irrespective of the other variable's value. They will intersect at a point where \( x \) is -8 and \( y \) is 11; thus, the point of intersection is \( (-8, 11) \).

Identifying the intersection of lines is particularly important because it can represent the solution to a system of equations, a point of equilibrium in economics, or even a location in navigation. In essence, it's a fundamental concept in various real-world applications involving graphical analysis.

Plotting Points

Plotting points is an essential skill in graphing. It's the process of marking a point on the coordinate plane where two values intersect. Each point represents a solution to an equation or a pair of values that relate to each other.

To plot a point, you start by identifying the x-coordinate and move along the x-axis to that location. Then, find the y-coordinate, and from your position on the x-axis, move vertically to the correct location on the y-axis. The spot where you end up is the point representing the ordered pair. For example, to plot the point \( (-8, 11) \), find -8 on the x-axis and then move up to 11 on the y-axis; place your point where these two meet.

Remember, a clear and correct representation on the graph leads to better understanding. Always ensure that each axis is neatly marked and that you move accurately along the axes to avoid errors. Practice plotting points as it's not just about drawing; it's about interpreting and understanding the relationship between variables in a visual and interactive way.