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The substitution method is a technique for solving a system of equations. It involves isolating one variable in one of the equations and then substituting that expression into the other equation. By doing this, we can find the value of one variable and then use it to find the other. This method is particularly useful when one of the equations is easy to solve for one variable.
Let's go through the Substitution method with an example:
Consider the system of equations:
1)  3x + y = -3
2)  x + 2y = 4
First, solve equation (1) for y:
\(y = -3 - 3x\)
Now, substitute this expression for y into equation (2):
\(x + 2(-3 - 3x) = 4\)
Simplify the equation:
\(x - 6 - 6x = 4\)
\(-5x - 6 = 4\)
Isolate x:
\(-5x = 10\)
\(x = -2\)
Now substitute x = -2 back into the equation for y:
\(y = -3 - 3(-2)\)
\(y = -3 + 6\)
\(y = 3\)
So, the solution is \((-2, 3)\). This simple example shows how the Substitution method can simplify solving systems of linear equations by reducing them to single-variable equations.

Solution verification is the process of confirming that the proposed solution satisfies both equations in a system. This step ensures that our solution is correct and reliable. Let's verify if (-2, 3) is a valid solution for the given system of equations:
The given system is:
1)  3x + y = -3
2)  x + 2y = 4
Step 1: Substitute x = -2 and y = 3 into the first equation:
\(3(-2) + 3 = -6 + 3 = -3\)
Since the left-hand side equals the right-hand side, the first equation is satisfied.
Step 2: Substitute x = -2 and y = 3 into the second equation:
\(-2 + 2(3) = -2 + 6 = 4\)
Again, since the left-hand side equals the right-hand side, the second equation is also satisfied.
Since both equations are satisfied, we confirm that (-2, 3) is indeed a solution for the system.
To understand why (3, -2) is not a solution, let's check both equations:
Substitute x = 3 and y = -2 into the first equation:
\(3(3) - 2 = 9 - 2 = 7\)
Since 7 does not equal -3, the first equation is not satisfied, confirming that (3, -2) is not a valid solution.
Verification is crucial because it ensures that the calculated values are accurate and meet all given conditions.

Linear equations are algebraic equations in which each term is either a constant or the product of a constant and a single variable. These equations graph as straight lines when plotted on a Cartesian plane. A system of linear equations includes two or more linear equations involving the same set of variables.
Here's a simple example of a system of linear equations:
1)  3x + y = -3
2)  x + 2y = 4
Each equation represents a straight line. The solution to the system is the point where both lines intersect. For linear equations with two variables, this point of intersection is found where the values for x and y satisfy both equations.
Linear equations are foundational in algebra because they help to understand more complex mathematical concepts. They are used in various fields such as physics, economics, and engineering to model different phenomena.
Understanding systems of linear equations is particularly useful for:

• Solving problems involving multiple constraints or conditions
• Modeling real-life situations such as supply and demand
• Analysis and optimization in operations research

Mastering linear equations and understanding how to solve systems of them using methods like substitution and verification are essential skills in mathematics and related disciplines.