Problem 56 - Algebra A Combined Function

Question

Use the system of linear equations below to answer the questions. \(\left\\{\begin{array}{l}x+y=4 \\ 2 x+b y=8\end{array}\right.\) a. Find the value of \(b\) so that the system has an infinite number of solutions. b. Find a value of \(b\) so that the system has a single solution.

Step-by-Step Solution

Answer

a. The value of \(b\) so that the system has infinite solutions is \(b = 2\). b. Any \(b \neq 2\) will result in a single solution.

Step 1: Understand the conditions for a system of linear equations

For a system of equations to have an infinite number of solutions, the equations must be dependent, meaning they are essentially the same line. This requires that coefficients are proportional. To have a single solution, the lines represented by the equations must intersect at exactly one point.

Step 2: Write the equations in standard form

Write the given system of equations in the form of:\[ a_1x + b_1y = c_1 \]\[ a_2x + b_2y = c_2 \] Thus we have:\[ 1x + 1y = 4 \] (Equation 1)\[ 2x + by = 8 \] (Equation 2).

Step 3: Condition for infinite solutions

For the system to have an infinite number of solutions, the second equation must be a multiple of the first. This requires: \[ \frac{2}{1} = \frac{b}{1} = \frac{8}{4} \]Solving \(b\) from: \[ \frac{2}{1} = \frac{b}{1} \], yields \(b = 2\).

Step 4: Verify condition for infinite solutions

Substitute \(b = 2\) in the second equation and check if both equations become multiples: \[ x + y = 4 \]\[ 2x + 2y = 8 \] Both equations represent the same line, confirming many solutions for \(b = 2\).

Step 5: Condition for a single solution

For the system to have a single solution, the equations must intersect at one point and therefore not be proportional. Any \(b eq 2\) ensures no proportionality.

Key Concepts

Infinite Solutions

Infinite solutions in a system of linear equations occur when both equations represent the same line. This means that for every point on one line, there's a corresponding point on the other line. This happens when the equations are dependent, meaning they share all possible solutions.

To check if a system of equations has infinite solutions, we look at their coefficients. Specifically, if the coefficients of the variables in both equations can be expressed as multiples of each other, the lines fall on top of each other and thus have infinite points of intersection.

In the given system:

• First Equation: \(x + y = 4\)
• Second Equation: \(2x + by = 8\)

To find infinite solutions, the second equation must be a multiple of the first, meaning their coefficients must be proportional. Solving \(\frac{2}{1} = \frac{b}{1} = \frac{8}{4}\), we conclude that when \(b = 2\), both equations describe the same line.

Single Solution

A system of linear equations that intersects at a single point has exactly one solution. This means that the lines represented by the equations must intersect at precisely one point on the graph.

For this to occur, the equations must be independent, meaning they are not multiples of each other. Simply put, if the slopes of the equations are different, they must intersect at a unique point, thus resulting in a single solution for the system.

With the given system:

• First Equation: \(x + y = 4\)
• Second Equation: \(2x + by = 8\)

The test for a single solution is ensuring the equations are not proportional. Thus, if \(b eq 2\), the lines will not be parallel or coincident, guaranteeing they meet at only one point.

Equations in Standard Form

Equations in standard form are commonly represented as \(ax + by = c\), where \(a\), \(b\), and \(c\) are integers. This form is very useful for analyzing systems of equations because it makes it easier to compare coefficients and determine relationships between lines.

In standard form:

• The coefficients \(a\) and \(b\) provide insight into the slope and direction of the line.
• The constant \(c\) helps in determining the line's position relative to the origin.

When studying systems, converting equations to standard form allows for a straightforward comparison. This makes it convenient to determine if two lines are parallel, if they intersect, or if they're coincident. In our exercise:

• Equation 1 is \(x + y = 4\)
• Equation 2 is \(2x + by = 8\)

Both express the classic structure of standard form, allowing easy examination of the relationships and solving for specific cases of solutions.