Problem 45 - 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 . \(\left\\{\begin{array}{l}x=5 \\ y=-2\end{array}\right.\)

Step-by-Step Solution

Answer

The lines intersect at a single point (5, -2), so the system has one solution.

Step 1: Interpret the given equations

We have the equations:1. \( x = 5 \)2. \( y = -2 \)These represent vertical and horizontal lines, respectively, in the coordinate plane.

Step 2: Analyze the nature of each line

The equation \( x = 5 \) represents a vertical line that crosses the x-axis at 5. The equation \( y = -2 \) represents a horizontal line that crosses the y-axis at -2.

Step 3: Determine if the lines intersect

Vertical lines always run parallel to the y-axis, while horizontal lines run parallel to the x-axis. They intersect at a single point where both equations are satisfied: \((x, y) = (5, -2)\).

Step 4: Conclusion on the type of lines

Since one line is vertical and the other is horizontal, they are neither identical nor parallel. They intersect at exactly one point.

Step 5: Conclusion on the number of solutions

The intersection point \((5, -2)\) is the only solution to the system of equations, which means the system has exactly one solution.

Key Concepts

Vertical Line

A vertical line in a coordinate system is a fascinating concept that often puzzles students at first. When you see the equation of a vertical line, it might look like this: \(x = a\), where \(a\) is a constant.
This equation tells us something very specific about every point on that line:

• Every point on the line has the same \(x\)-coordinate value, \(a\).
• The line runs parallel to the y-axis.

This type of line doesn't "tilt" or "lean." It goes straight up and down, cutting the x-axis at the point \((a, 0)\).
Understanding vertical lines is crucial because they behave differently than sloped lines or horizontal lines:

• They can't be defined in terms of a slope, as the slope is actually undefined (with an infinite rise over zero run).
• They will intersect horizontal lines and only intersect with another vertical line if both have the same \(x\)-coordinate value.

The equation \(x = 5\) represents a vertical line that crosses the x-axis at 5, meaning every point on this line has an \(x\) value of 5.

Horizontal Line

Horizontal lines are a bit more intuitive. When you see an equation like \(y = b\), it indicates a horizontal line.
Here's what you need to know about horizontal lines:

• Every point on this line has the same \(y\)-coordinate value, \(b\).
• They run parallel to the x-axis.

This means a horizontal line is perfectly flat with no slope, as it doesn't rise or fall; the slope is zero.
Horizontal lines are often simpler to understand because they reflect real-world scenarios, like a floor or a ceiling.
The equation \(y = -2\) means the line crosses the y-axis at \(-2\), and every point on this line has a \(y\) value of \(-2\).
Horizontal lines will intersect with vertical lines, but they are parallel to another horizontal line if both have the same \(y\)-coordinate value.

Systems of Equations

Systems of equations directly deal with multiple equations working together. The goal is to find common solutions that satisfy all the equations simultaneously.
This particular exercise involves a system with one vertical line \(x = 5\) and one horizontal line \(y = -2\).
The two lines form a special scenario:

• These lines intersect at precisely one point, meaning they share exactly one set of \((x, y)\) coordinates that satisfy both equations simultaneously.
• This intersection point is where both lines meet, and no other points share this characteristic for such a system of equations.

In a broad sense, systems of equations can have:

• A single solution if they intersect at a point like in this example.
• Infinitely many solutions if they are identical lines.
• No solutions if they are parallel (and not identical).

By realizing how each type of line interacts in a system, you'll be able to visualize and solve complex equations more intuitively.