Problem 60 - Algebra 1

Question

Graph the two lines in the same coordinate plane. Then find the coordinates of the point at which the lines cross. \(x=-5, y=2\)

Step-by-Step Solution

Answer

The coordinates of the point at which the two lines intersect are (-5, 2).

Step 1: Analyze the given lines

Notice that the lines \(x=-5\) and \(y=2\) are not the typical form of linear equations (y = mx + b). Both equations only contain one variable. \(x=-5\) is a vertical line passing through the point at -5 on the x-axis. \(y=2\) is a horizontal line passing through the point at 2 on the y-axis. This distinction becomes important when sketching the graphs.

Step 2: Graphing the lines on the same coordinate plane

Start to draw a Cartesian plane (a plain with x and y axes). The first line to graph is \(x=-5\). This is a vertical line passing through the point -5 on the x-axis. The second line, \(y=2\), is a horizontal line passing through the point 2 on the y-axis. Draw these on the same plane.

Step 3: Identify the intersection

The two lines intersect at the point where \(x=-5\) and \(y=2\). Therefore, the point of intersection is (-5, 2)

Key Concepts

Coordinate Plane

A coordinate plane, also known as a Cartesian plane, is a two-dimensional plane formed by the intersection of two perpendicular lines: the x-axis and the y-axis. These axes divide the plane into four quadrants, each helping us to locate points in 2D space through coordinates, typically in the format \(x, y\).

• The x-axis is a horizontal line that runs left to right.
• The y-axis is a vertical line that runs top to bottom.
• The point where these axes intersect is called the origin, represented as (0,0).

Coordinates allow us to graphically represent solutions to equations like lines, curves, and shapes.
By understanding how to place points and draw lines on a coordinate plane, we establish a groundwork for studying geometry and algebraic concepts.

Vertical and Horizontal Lines

Vertical and horizontal lines are quite straightforward to identify and graph because they feature only one variable in their equations. Let's explore each:
**Vertical Lines**
Vertical lines have equations of the form \(x = a\). Here, the value of the x-coordinate is fixed at 'a' for all points on the line. No matter the y-value, the x-value remains constant.

• Example: \(x = -5\) is a vertical line crossing the x-axis at -5.
• These lines are parallel to the y-axis and have no slope (undefined slope).

**Horizontal Lines**
Horizontal lines are expressed with the equation \(y = b\), fixing the y-coordinate at 'b' while the x-value can vary freely.

• Example: \(y = 2\) is a horizontal line crossing the y-axis at 2.
• These lines run parallel to the x-axis and have a slope of 0.

Understanding these lines is essential for graphing purposes, especially when such lines are part of problem solutions or modeling real-world scenarios.

Intersection of Lines

The intersection of lines refers to the exact point where two lines on the coordinate plane meet or cross each other. This point can be crucial, especially in finding solutions to simultaneous equations.
In our given exercise, we have two lines: \(x = -5\) and \(y = 2\).

• The vertical line \(x = -5\) dictates that at this specific x-coordinate, the line extends infinitely in the y-direction.
• The horizontal line \(y = 2\) specifies that at the y-coordinate of 2, the line extends infinitely in the x-direction.

The point where these two lines intersect is where both conditions are satisfied simultaneously: \(x = -5\) and \(y = 2\). Therefore, the coordinates of their intersection are (-5, 2). This reveals the solution to any set of equations represented by these lines. Not only does this highlight their intersection, but it also shows how graphically solving equations can provide insights into mathematical relationships.