91Ó°ÊÓ

In algebra, a system of equations consists of two or more equations with the same set of variables. A dependent system occurs when the equations describe the same line in the coordinate plane. This means every solution of one equation is also a solution of the other. In other words, the equations are equivalent to each other.

To identify a dependent system, simplify both equations and see if one is a multiple of the other. For example, in this exercise, the equations are:

• \(4x - y = 9 \)
• \(-8x + 2y = -18 \)

When the second equation is divided by 2, it becomes \(-4x + y = -9\), which is the negative of the first equation.

This shows that both equations are actually representations of the same line, making the system dependent.

When a system of equations has infinitely many solutions, it implies that there are countless points that satisfy both equations. In geometric terms, this means the equations represent the same line and thus share every single point on that line.

To express the solutions, we can solve one of the equations for one variable in terms of the other. For instance, solving \(4x - y = 9 \) for x yields:

\(4x = y + 9\)

\(x = \frac{y + 9}{4}\)

Thus, any value for y can be substituted into this equation to find a corresponding x. This gives us a solution set in the form:
\((x, y) = (\frac{y + 9}{4}, y) \), where y is arbitrary.

This shows all possible solutions lie along the same line, proving there are infinitely many of them.

Linear equations are equations of the first degree, meaning they can be written in the form \(ax + by = c\), where a, b, and c are constants. In a system of linear equations, we solve for the variables x and y to find where their lines intersect.

A linear equation will graph as a straight line in the coordinate plane. When these linear equations create a system, their relationship can be one of the following:

• Independent (intersecting at a point)
• Dependent (the same line, infinite solutions)
• Inconsistent (parallel lines, no solutions)

In our example, the system:

• \(4x - y = 9\)
• \(-8x + 2y = -18\)

was found to be dependent, as already discussed. This implies that linear equations can sometimes be equivalent, leading to dependent systems with infinitely many solutions.