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Chapter 4: Problem 45

Without graphing, decide. See Examples 7 and \(8 .\) 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? $$ \left\\{\begin{array}{l} {x=5} \\ {y=-2} \end{array}\right. $$

Short Answer

Expert verified
The lines intersect at one point, so the system has one solution.

Step by step solution

01

Identify the Form of Each Equation

The given system consists of two equations: \( x = 5 \) and \( y = -2 \). Each equation represents a vertical or horizontal line. \( x = 5 \) is a vertical line passing through all points where \( x \) is 5. \( y = -2 \) is a horizontal line passing through all points where \( y \) is -2.
02

Determine Intersection Points

To find where the lines intersect, check for a point that satisfies both equations. The only point where both equations are true simultaneously is the point \( (5, -2) \).
03

Decide the Relationship Between the Lines

Since there is only one point where these two lines meet, the lines intersect at exactly one point. Thus, the lines are neither parallel nor identical but intersect at the point \( (5, -2) \).
04

Count the Number of Solutions

A system of equations has as many solutions as points where the graphs intersect. Since the lines intersect at exactly one point, the system has exactly one solution.

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Key Concepts

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

Graphing Linear Equations
Graphing linear equations is a fundamental skill in understanding systems of equations. When you graph a linear equation like \( x = 5 \), you are creating a vertical line on the coordinate plane. It means that for every value of \( y \), the \( x \)-coordinate remains constant at 5. Similarly, the equation \( y = -2 \) forms a horizontal line where every \( x \)-value corresponds to the same \( y \)-value of -2.

To graph these lines:
  • Vertical Lines: Simply draw a line that passes through the specified \( x \)-value, extending infinitely in the up and down directions. For \( x = 5 \), the line doesn't tilt or curve, but remains perfectly straight, intersecting the x-axis at 5.
  • Horizontal Lines: A horizontal line stays flat and level, passing through the given \( y \)-value. For \( y = -2 \), draw a line crossing the y-axis at -2, continuing indefinitely to the left and right.
The intersection of these specific lines helps us visualize their relationship, which leads to our next key concept.
Intersection of Lines
The intersection of lines is a critical concept in solving systems of equations. This intersection is the point where two graphs meet on a coordinate plane. It represents a solution common to both equations.

For the lines \( x = 5 \) and \( y = -2 \), their intersection is the single point \( (5, -2) \). This point is crucial because:
  • It satisfies both equations simultaneously, meaning both conditions for \( x \) and \( y \) are met here.
  • Since no other points satisfy both conditions, this intersection signifies the only solution of the system.
In systems with more complex equations, the intersection might be found by algebraic methods or visually on a graphing tool. Understanding how lines intersect can guide you in determining the solution type, such as whether the system has one solution, infinitely many, or none (when the lines are parallel).
Solutions to Equations
Solutions to equations depend on how their corresponding lines relate to each other on a graph. A solution is simply a point of intersection between the lines.

In our system of equations:
  • If the lines intersect at a single point, like \( (5, -2) \), there is exactly one solution. This indicates an independent system where the lines aren't parallel or coincident.
  • If the lines were coincident (identical), they would share infinitely many solutions because every point on the line is a solution.
  • If the lines were parallel, they would never meet, and thus the system would have no solutions.
Recognizing the number and type of solutions in a system of equations tells us a lot about their graphical relationship. It also informs us on whether or not the system is solvable or has multiple overlapping solutions.

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