91Ó°ÊÓ

A system of equations will have a unique solution when the two lines represented by the equations intersect at precisely one point on a graph. Imagine two straight lines crossing each other; this is where they meet only once. To check for a unique solution, we examine the slopes of both lines.

In our exercise, the equations were:

• Equation 1: \(kx + y = -2\)
• Equation 2: \(x - y = 2\)

For these lines to intersect at just one point, they must have different slopes. Recall that the slope-intercept form is \(y = mx + b\), where \(m\) represents the slope. By rearranging Equation 1 to this form, we get \(y = -kx - 2\), and from Equation 2, it simplifies to \(y = x - 2\).

To have different slopes:

• The ratio of the coefficients of \(x\) in both equations must not be the same; that is \(k/1 eq 1/-1\). Simplifying, we find that \(k eq -1\).

If \(k\) is anything other than \(-1\), the lines will intersect once, resulting in a unique solution.

When a system of equations has no solution, it means that the lines are parallel and never meet. Imagine two train tracks that run side by side but never cross. For this to hold true, the lines have the same slope but different y-intercepts. This parallel scenario means they are equidistant and will never touch.

For our equations, let's recall:

• Equation 1: \(kx + y = -2\)
• Equation 2: \(x - y = 2\)

To check for no solution, examine the coefficients. Lines will be parallel if the ratio of the coefficients of \(x\) in both equations equals the ratio of the coefficients of \(y\), but the constant terms do not match. Looking at the given system:

Compare the ratios:

• \(k/1 = 1/-1\)
• \(-2/(-2) eq 4/2\)

The condition \(k = -1\) holds for the coefficients, making the lines parallel. Since the constant terms don't match, the system has no solution when \(k = -1\).

Infinitely many solutions arise in a system of equations when the lines are essentially the same line, meaning they coincide perfectly on the graph. It's like saying one line is neatly stacked on top of another, so every point on one line is also on the other line.

In the provided system of equations, we are asked to find a scenario where this happens:

• Equation 1: \(kx + y = -2\)
• Equation 2: \(x - y = 2\)

For the lines to fully overlap, every part of Equation 1 must be a multiple of Equation 2.

This means:

• The coefficients of \(x\) and \(y\) must be proportional as well as the constants on the other side:
• \(k/1 = 1/-1\)
• \(-2/(-2) = 4/2\)

However, the constants do not satisfy the condition of equality, meaning that the lines will never completely overlap. Thus, in this system, there is no value of \(k\) where the system has infinitely many solutions.