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Linear equations form the backbone of algebra and are fundamentally equations of the first degree. This means that variables within the equation are not raised to any power other than one. To solve linear equations, one must find the value of the unknown variables that make the equation true. A robust understanding of basic arithmetic operations - addition, subtraction, multiplication, and division - is essential because these operations are used to isolate the variable.

For example, consider the equation from our exercise, where we have two separate equations for the variable y. The essence of solving for x is to make the equations equal to each other because they intersect at a common point. By manipulating the equations, as seen in steps 1 and 2, we remove the variable y and solve for x using arithmetic operations that maintain the balance of the equation.

One practical approach to solving systems of linear equations is the substitution method. This method involves taking the solution from one equation and substituting it into another to find the remaining unknowns. The main goal is to reduce the system of equations to one equation with one variable.

As demonstrated in the provided exercise, once x is found, that value is substituted back into one of the original equations. Efficient use of substitution can quickly lead to the solution, as shown in step 3. An important aspect of substitution is choosing the most straightforward equation to substitute into to minimize computational errors and complexity.

Algebraic equations are equations that include numbers, operations, and variables. They represent relationships between quantities where some quantities are unknown. These equations can vary in complexity from simple linear equations to more intricate polynomial equations. While the focus of the exercise is on linear equations, it's part of a broader category of algebraic expressions.

Particularly, in the context of our exercise, recognizing that both given equations describe a line in a coordinate system is vital. Their intersection is the solution to the system. Grasping the properties and solving techniques of algebraic equations enables students to navigate through more complex algebraic concepts.