Solve using the elimination method. Also determine whether...

Solve using the elimination method. Also determine whether each system is consistent or inconsistent and whether the equations are dependent or independent. $\frac{1}{5} x+\frac{1}{2} y=6$ $\frac{3}{5} x-\frac{1}{2} y=2$ (Hint: First multiply by the least common denominator to clear fractions.)

Solve using the elimination method. Also determine whether each system is consistent or inconsistent and whether the equations are dependent or independent.
$\frac{1}{5} x+\frac{1}{2} y=6$
$\frac{3}{5} x-\frac{1}{2} y=2$
(Hint: First multiply by the least common denominator to clear fractions.)

Key Concepts

Clearing Fractions

Clearing fractions is a preprocessing technique used in solving equations; it involves multiplying every term by the least common denominator to eliminate fractional coefficients. This step simplifies the arithmetic and makes further processes such as elimination or substitution more straightforward.

Consistent vs Inconsistent Systems

A consistent system is one where at least one solution exists for the set of equations, meaning the equations intersect at one point or are coincident. An inconsistent system, on the other hand, has no solution because the equations describe parallel lines that never intersect.

Dependent vs Independent Systems

Dependent systems are those where one equation is a multiple of the other, leading to an infinite number of solutions since both equations represent the same line. In contrast, independent systems consist of distinct equations that intersect at a single point, which results in a unique solution.

Elimination Method

The elimination method is an algebraic technique used to solve systems of linear equations. It involves aligning the equations so that adding or subtracting them will cancel out one of the variables, allowing for the remaining variable to be solved easily. This method is especially useful for systems where direct substitution is cumbersome.

Systems of Linear Equations

A system of linear equations includes two or more equations that have variables in common. The goal is to find values for these variables that satisfy each equation simultaneously. Such systems can have one solution, infinitely many solutions, or no solution at all, depending on how the equations relate to each other.

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