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Chapter 4: Problem 55

Solve by graphing. $$ \\{y=5 y=-5 $$

Short Answer

Expert verified
The equations do not intersect; thus, there is no solution.

Step by step solution

01

Understand the Problem

We have two equations: \( y = 5 \) and \( y = -5 \). These are horizontal lines on a graph, where \( y \) is constant.
02

Graph the First Equation

Plot the first equation \( y = 5 \) on the graph. This is a horizontal line that passes through all points where the y-coordinate is 5.
03

Graph the Second Equation

Plot the second equation \( y = -5 \) on the graph. This is a horizontal line that passes through all points where the y-coordinate is -5.
04

Analyze the Intersection

Since both lines are horizontal and have different y-values, they will never intersect. This means there is no solution, as there is no point that satisfies both equations simultaneously.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Horizontal Lines
Horizontal lines on a graph are lines where the y-coordinate remains consistent across all x-values. This means the line runs parallel to the x-axis. For instance, given the equation \( y = 5 \), every point on this line has a y-coordinate of 5, such as \((1, 5)\), \((2, 5)\), and \((-3, 5)\). This creates a straight line that is horizontal. It does not slope upward or downward. The simplicity of horizontal lines makes them easy to graph once you understand that only the y-value affects the equation and all x-values are included.
Intersection of Lines
When dealing with the intersection of lines, we look for a point that both lines pass through. For lines to intersect, they must share at least one common point. In graphing systems of equations, this point of intersection represents the solution to the system. If two lines intersect, it usually means their equations can be solved together. However, if lines are parallel, like horizontal lines with different y-values, they do not intersect, indicating no shared solution. If two lines overlap completely, they have infinitely many intersection points, meaning infinitely many solutions.
No Solution in Systems of Equations
Sometimes, systems of equations do not have any solutions. This happens when the graphs of the equations do not intersect. In the case of the exercise \( y = 5 \) and \( y = -5 \), both equations are horizontal lines but on different levels of the y-axis. Since they have different y-values and run parallel to the x-axis, they never meet. No shared point means no solution for the system. When graphing, no intersection emphasizes that there is no set of values for x and y that can satisfy both equations simultaneously. Understanding this concept is crucial in recognizing the outcomes when dealing with linear systems.

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