Problem 48 - Elementary Algebra Within Reach

Question

Consider the system of linear equations. $$ \left\\{\begin{array}{r} x+y=8 \\ 2 x+2 y=k \end{array}\right. $$ (a) Find the value(s) of \(k\) for which the system has an infinite number of solutions. (b) Find one value of \(k\) for which the system has no solution. (There are many correct answers.) (c) Can the system have a single solution for some value of \(k\) ? Why or why not?

Step-by-Step Solution

Answer

The system of equations has an infinite number of solutions when k = 16. It has no solution for values of k different from 16, such as k = 20. This system cannot have a single solution for any value of k.

Step 1: Identify Infinite Solutions

For a system of equations to have infinite solutions, the equations must be identical. Therefore, the second equation must be a multiple of the first. Simplify the second equation by dividing every term by 2 to match it with the first equation: 2x + 2y = k becomes x + y = k/2. For the equations to be the same, k/2 must equal 8, so k must be 16. Therefore, the value of k for which the system has an infinite number of solutions is 16.

Step 2: Identify No Solutions

For a system of equations to have no solutions, the equations must represent parallel lines. In this case, the equations have the same coefficients for the x and y terms, so they could be parallel lines if the constants are different. Any value of k other than 16 will make the second equation different from the first, so there are many correct answers. One such value could be k = 20.

Step 3: Identify Single Solution

For a system of equations to have a single solution, the two lines represented by the equations must intersect at one point. However, the coefficients of the x and y terms in these two equations are the same, making them either the same line (infinite solutions) or parallel lines (no solutions), depending on the value of k. So, this system cannot have a single solution for any value of k.

Key Concepts

Understanding Infinite Solutions

A system of linear equations has infinite solutions when every point on the line described by one equation is also on the line described by the other. This occurs when the equations are essentially the same, or in other words, when they are multiples of each other.
For example, consider the equations:

• \(x + y = 8\)
• \(2x + 2y = k\)

To have infinite solutions, the second equation must be a scalar multiple of the first, meaning every term in the second equation should be a scaled version of the first equation's terms. When simplified, the second equation becomes \(x + y = \frac{k}{2}\). For these equations to be identical, \(\frac{k}{2} = 8\), leading to \(k = 16\). This value of \(k\) ensures that both equations describe the exact same line, resulting in an infinite number of solutions.

Parallel Lines and No Solutions

Parallel lines are like train tracks extending into infinity, never meeting. They occur when systems of equations have the same coefficients for their variables, signifying equal slopes, but different constant terms.
With parallel lines, the system has no solutions because they never intersect anywhere on a graph. For our example:

• \(x + y = 8\)
• \(2x + 2y = k\)

Simplifying the second equation gives us \(x + y = \frac{k}{2}\). For the lines to be parallel and not the same, \(\frac{k}{2} eq 8\), meaning \(k\) must not equal 16. Any number other than 16 for \(k\) results in parallel lines where no intersection point exists. Thus, one possible value for \(k\) could be 20 as an example of a scenario with no solutions.

Finding an Intersection Point

An intersection point happens when two lines on a graph meet at a single point. This implies the system of equations has a unique solution. However, for two lines to intersect, they must have different slopes.
In this specific case, the equations:

• \(x + y = 8\)
• \(2x + 2y = k\)

After reducing, both equations can become lines with the same slope due to identical x and y coefficients. This results in either identical lines (infinite solutions) or parallel lines (no solutions). Since identical slopes cannot create different lines that intersect once, there will never be a value for \(k\) that results in a single intersection point for this system.