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A system of equations is a set of two or more equations that you deal with together at once. These equations have common variables, and the aim is to find the values of these variables that satisfy all the equations simultaneously. Such systems can appear in various forms, with linear equations being the most basic type, involving expressions where each term is either a constant or the product of a constant and a single variable.

In our example, we have:

• Equation 1: \(5x - 2y = -3\)
• Equation 2: \(3x - y = 1\)

The primary goal is to find the values of \(x\) and \(y\) that make both equations true. This shared solution shows where the graphs of the equations intersect when plotted on a coordinate plane. Solving these systems requires skill in manipulating the equations using various methods, like substitution or elimination.

The elimination method is a popular technique for solving systems of linear equations. Its main objective is to eliminate one of the variables by adding or subtracting the equations, making it easier to solve for the remaining variable.

To use elimination effectively, you should:

• Ensure that the coefficients for one of the variables are equal or can be made equal. In our exercise, that's done by multiplying the second equation by 2, resulting in matching coefficients for \(y\).
• Once the coefficients match, you can add or subtract one equation from the other to remove a variable. Here, subtracting the modified second equation from the first eliminates \(y\).
• Solve the resulting simpler equation to find one of the variables. For instance, \(-x = -5\) simplifies to \(x = 5\).
• Substitute the known value back into one of the original equations to find the other variable.

This method is particularly useful when the equations are easily manipulated to have equal coefficients, enabling a straightforward solution.

After solving a system of equations, it is crucial to verify that the solution satisfies both original equations. Verification ensures that no errors were made during the calculation process. Here's how:

1. Substitute the solution back into each original equation:

• For \((x, y) = (5, 14)\), substitute \(x = 5\) and \(y = 14\) back into the first equation: \(5(5) - 2(14) = 25 - 28 = -3\), which holds true.
• Then check the second equation: \(3(5) - 14 = 15 - 14 = 1\), confirming its accuracy.

By confirming the solution provides true results for both equations, you validate its correctness. Such checks are essential, as minor errors can lead to incorrect answers. Always take a moment to verify, ensuring the solution accurately addresses the system.