Problem 46 - Algebra A Combined Function

Question

Without graphing, decide. a. Are the graphs of the equations identical lines, parallel lines, or lines intersecting at a single point? b. How many solutions does the system have? See Examples 7 and 8 . \(\left\\{\begin{array}{l}y=3 \\ x=-4\end{array}\right.\)

Step-by-Step Solution

Answer

The lines intersect at one point and the system has one solution.

Step 1: Recognize the Equations

We are given two equations: \( y = 3 \) and \( x = -4 \). These equations represent vertical and horizontal lines.

Step 2: Analyze Geometry

The equation \( y = 3 \) represents a horizontal line crossing the y-axis at 3, and the equation \( x = -4 \) represents a vertical line crossing the x-axis at -4.

Step 3: Identify Intersections

A horizontal line \( y = 3 \) and a vertical line \( x = -4 \) intersect at exactly one point, the point \((-4, 3)\).

Step 4: Conclusion on Graph Type

The lines are neither identical nor parallel. They are perpendicular, crossing at the point \((-4, 3)\).

Step 5: Determine Number of Solutions

Since the lines intersect at a single point, the system has exactly one solution.

Key Concepts

Intersection Point

An intersection point is where two lines meet or cross each other. In a system of linear equations, this point is crucial as it represents the common solution. For example, consider the system given by the equations \( y = 3 \) and \( x = -4 \). The first equation signifies a horizontal line, and the second, a vertical line. These lines intersect because one runs side to side, while the other runs up and down, meeting at right angles. The intersection point of these lines is found by combining their properties: the horizontal line passes through all points where \( y = 3 \) and the vertical line passes through all points where \( x = -4 \). Hence, their intersection occurs at the point \((-4, 3)\).

• This point is unique because it is the only location where both conditions \( y = 3 \) and \( x = -4 \) are simultaneously satisfied.
• Every system of linear equations will either have one intersection point, infinitely many intersection points, or no intersection points at all.

Understanding how to determine the intersection point of a system of equations helps to find solutions without necessarily having to graph the equations.

Horizontal and Vertical Lines

Horizontal and vertical lines have specific characteristics that make them straightforward to work with in algebra. A **horizontal line** on a graph is perfectly flat, running parallel to the x-axis. The equation of a horizontal line is of the form \( y = c \), where \( c \) is a constant. This tells you the line crosses the y-axis at this point, and the y-value remains consistent along its length.

• For example, the line described by \( y = 3 \) is horizontal and passes through every point where the y-coordinate is 3.
• Its slope is zero because it doesn’t rise vertically when moving from left to right.

In contrast, a **vertical line** runs straight up and down, parallel to the y-axis. Its equation is \( x = k \), where \( k \) is a constant that signifies where the line crosses the x-axis.

• The line given by \( x = -4 \) is vertical, intersecting the x-axis at -4, maintaining an unchanged x-value along its length.
• Vertical lines have an undefined slope because any movement along the line involves infinite vertical change with no horizontal change.

These simple yet distinct lines are critical in identifying the nature of intersections within systems of linear equations.

Number of Solutions in a System

The number of solutions in a system of linear equations indicates how many points satisfy both equations simultaneously. For systems represented by two lines on a graph, the possibilities are:

• **One solution**: This occurs when the lines intersect at exactly one point, indicating the system's equations meet only once. For instance, our system with equations \( y = 3 \) and \( x = -4 \) intersects once at the point \((-4, 3)\).
• **No solution**: When lines are parallel, they never meet and thus have no common point of intersection. These lines have identical slopes but different y-intercepts.
• **Infinitely many solutions**: If the lines are identical, then each point on one line is a point on the other as well. This means every point is a solution to the system.

Understanding the number of solutions helps in determining the relationship between the equations:- **Intersection Point and Solution Count**: Lines meeting at one point have a unique solution for that system.- **Parallel Lines**: Indicate no solutions, as the equations describe distinct paths.- **Identical Lines**: Reveal a complete overlap in solutions.