Problem 43 - 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 . \(\left\\{\begin{array}{l}4 x-y=6 \\ \frac{1}{2} y=-3+2 x\end{array}\right.\)

Step-by-Step Solution

Answer

The graphs represent identical lines with infinitely many solutions.

Step 1: Simplify the Second Equation

The second equation is given as \( \frac{1}{2} y = -3 + 2x \). To simplify, multiply both sides by 2 to eliminate the fraction: \( y = -6 + 4x \).

Step 2: Rearrange the First Equation

The first equation is originally \( 4x - y = 6 \). Rearrange it to isolate \( y \) on one side: \( y = 4x - 6 \).

Step 3: Compare the Two Equations

After simplification, both equations are \( y = 4x - 6 \). Since the equations are identical, they represent the same line.

Step 4: Determine the Type of Lines

Since the two equations are equivalent, they correspond to identical lines.

Step 5: Count the Number of Solutions

Identical lines indicate that every point on the line is a solution, thus the system has infinitely many solutions.

Key Concepts

Systems of Equations

A system of equations is precisely that – a collection of two or more equations with a common set of variables. In instances where you are dealing with linear equations, which are equations of form like \(ax + by = c\), these systems are often depicted as lines on a graph.
The main goal when working with systems of equations is to find the set of values for the variables that satisfy all the given equations simultaneously. The solution to a system of equations can either be:

• A single point where the lines intersect.
• No solution if the lines are parallel and never meet.
• Infinitely many solutions if the lines coincide, meaning they are the same line.

This problem-solving technique is instrumental in fields like engineering and economics, where multi-variable problems are prevalent.
Understanding and solving systems of equations will greatly enhance your mathematical problem-solving skills.

Identical Lines

Identical lines in the context of systems of equations imply that the two lines are the same. This means every point on one line is exactly the same as every point on the other line.
In mathematical terms, this occurs when two linear equations are actually different forms of the same equation. For example, in the original problem, after simplifying both equations, both were rearranged into the same form: \(y = 4x - 6\).
Thus, they describe the identical line in a coordinate plane. Understanding identical lines is crucial because it helps us determine whether two functions behave the same way, which means their graphs are the same.

Infinitely Many Solutions

When dealing with systems of equations, one possible outcome is having infinitely many solutions. This occurs when the equations in the system are identical or equivalent, resulting in the same line when graphed.
Since every point on one line corresponds to a point on the other, there’s no distinct single solution. Instead, every coordinate on the line counts as a solution to the system.

• In simple terms, if two equations are versions of the same line, then they intersect at every point, meaning infinitely many solutions.
• This contrasts sharply with situations where lines meet at a single point, giving exactly one solution, or are parallel with no solutions.

Grasping the concept of infinitely many solutions is crucial because it highlights deeper relationships between equations in algebraic systems.