Problem 52 - 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 . \(\left\\{\begin{array}{l}2 x+y=0 \\ y=-2 x+1\end{array}\right.\)

Step-by-Step Solution

Answer

The lines are parallel, so the system has no solutions.

Step 1: Rewrite in standard form

The first equation is already in standard form: \(2x + y = 0\). For the second equation, rewrite it as \(2x + y = 1\). Now, both equations are in the form \(Ax + By = C\).

Step 2: Compare coefficients

The coefficients of \(x\) and \(y\) in both equations are 2 and 1, respectively, which are the same. However, the constant terms \(C\) are 0 and 1, which are different.

Step 3: Determine relationship from coefficients

Since the coefficients of the variables are the same but the constants are different, the lines represented by these equations are parallel. This means they do not intersect.

Step 4: Determine number of solutions

Since the lines are parallel and do not intersect at any point, the system of equations has no solutions.

Key Concepts

Parallel Lines

In the world of geometry, parallel lines are lines in a plane that do not meet. They are equidistant from each other at every point, which means that no matter how far they are extended, they will never touch. In the context of linear equations, two lines are parallel if they have the same slope but different y-intercepts.
When we talk about slopes in linear equations, they reflect the steepness and direction of the lines. If two lines have identical slopes, they will sit parallel to each other in a graph.

• Example: Suppose we have two lines described by the equations \(y = 2x + 3\) and \(y = 2x - 4\). Here, both lines have the slope of 2, indicating they are parallel.

In summary, recognizing parallelism in lines involves examining their slopes. Identical slopes lead to parallel lines.

Solution of a System

In mathematics, a solution to a system of equations is a set of values for the variables that satisfy all equations simultaneously. This means that by substituting the values into each equation, they hold true. When working with systems of linear equations, the behavior of the lines involved often dictates the nature and amount of solutions possible.

• If the lines intersect at exactly one point, there is a unique solution, as this is the only set of values that satisfies each equation.
• If the lines are parallel, then there are no solutions, as they never intersect.
• If the lines are identical, every point on the line is a solution, making for infinite solutions.

Understanding the relationship between the lines thus becomes crucial in determining how many solutions a system holds.

Standard Form of a Linear Equation

The standard form of a linear equation is a way of organizing linear equations in a linear manner that makes them easy to read and analyze. It is customarily expressed as \(Ax + By = C\), where \(A\), \(B\), and \(C\) are integers, and \(A\) and \(B\) are not both zero.
This form is beneficial because it readily displays the coefficients which can be compared across multiple equations to deduce relationships between the lines.

• Example: The equation \(3x + 4y = 12\) is in standard form, with coefficients \(A = 3\), \(B = 4\), and \(C = 12\).
• All terms are aligned to help easily compare between different equations based on their coefficients.

Using this convenient format, one can quickly assess important characteristics such as parallelism or intersection points by examining or manipulating the equations.

Coefficients Comparison

When analyzing systems of linear equations, comparison of coefficients plays a crucial role. This involves comparing the coefficients of the variables in each equation to understand their relationship.
In a standard form equation \(Ax + By = C\), coefficients \(A\) and \(B\) denote the "weights" of the variables. By examining these, one can determine whether lines are parallel, identical, or intersecting at a single point.

• Parallel lines have identical coefficients for \(x\) and \(y\), but different constant terms \(C\).
• Identical lines have identical coefficients for both the variables and the constant terms.

Thus, by leveraging coefficients comparison, valuable insights into the geometric relationship between lines can be gleaned without the necessity of graphical representation.