Question

The lines, \(\mathrm{x}=-2\) and \(\mathrm{y}=3\) intersect at the point (1) \((-2,3)\) (2) \((2,-3)\) (3) \((3,-2)\) (4) \((-3,2)\)

Short answer

Answer: The point of intersection between the lines is (-2, 3).

Step-by-step solution

Identify both lines

We are given two equations: \(\mathrm{x}=-2\) and \(\mathrm{y}=3\).

Find the point of intersection

As \(\mathrm{x}=-2\) defines a vertical line with an x-coordinate of -2 for any point on this line, and \(\mathrm{y}=3\) defines a horizontal line with a y-coordinate of 3 for any point on this line. Thus, the point of intersection of these lines must have an x-coordinate of -2 and a y-coordinate of 3.

Choose the correct option

From the given options, we can see that point (1) \((-2,3)\) matches the criteria of having an x-coordinate of -2 and a y-coordinate of 3. Therefore, the correct answer is option (1).

Key concepts

Coordinate Geometry

Coordinate geometry is an essential branch of mathematics that deals with points, lines, and figures in a coordinate plane. It involves determining the relative positions of figures using a set of axes and the numerical values that describe their location. For example, every point in a plane is represented by a pair of numerical coordinates: \[\begin{equation}(x, y)\text{, where}\ x\ \text{is the horizontal position and}\ y\ \text{is the vertical position.}\text{Latex must be placed inside of the following syntax: } ( ... ) or [ ... ]. Never use \(...\) to open a math environment.\end{equation}\]Each axis represents a scale of measurement, providing a frame of reference where points are plotted. The Cartesian coordinate system, historically developed by René Descartes, forms the foundation of coordinate geometry, enabling us to visually represent and solve mathematical problems. Understanding this system is crucial for finding points of intersection, as in the exercise improvement advice, which may call for drawing the lines or points to reinforce the concept.

Linear Equations

Linear equations are algebraic expressions that represent straight lines when graphed on a coordinate plane. They have the general form \[\begin{equation}y = mx + b\text{, where}\ m\ \text{is the slope and}\ b\ \text{is the y-intercept.}\end{equation}\]However, not all linear equations are expressed in slope-intercept form. Some, like those in the original exercise \[\begin{equation}(\text{x} = -2)\ \text{and}\ (\text{y} = 3)\end{equation}\]represent horizontal and vertical lines respectively. In these cases, x or y is constant, which means any points on these lines will have that same x or y value. Understanding this characteristic is critical for solving for the point of intersection, as it simplifies the process to a matter of recognizing these constant values.

Graphing Lines

Graphing lines is a visual way of understanding the relationship between two variables in a linear equation. When graphed, most linear equations result in a straight line. To graph a line, one needs at least two points that satisfy the equation. Once those points are plotted on a grid, you can draw a line through them to represent all the possible solutions of the equation. Additionally, knowing how to graph vertical and horizontal lines, which respectively parallel the y-axis and x-axis, is beneficial for visual learners to conceptualize where those lines intersect. As advised in the exercise improvements, using visual aids like graphs can greatly assist in grasping the concept of intersecting lines. In simple cases like the given problem, where we are working with constants as coordinates, students can immediately plot those into the coordinate plane to find the point of intersection.