91Ó°ÊÓ

A system of linear equations consists of two or more linear equations with the same set of variables. In our case, we have a system with two linear equations:

• \( x + 2y = -1 \)
• \( 2x - 3y = 12 \)

The solution to a system of linear equations is the set of values for the variables that satisfy all equations in the system simultaneously. Generally, these systems can have:

• One unique solution, if the equations intersect at a single point.
• No solution, if the equations represent parallel lines.
• Infinitely many solutions, if the equations are identical, representing the same line.

In our exercise, the goal is to find the unique values of \( x \) and \( y \) using the substitution method, which we'll explain in detail in the following sections.

Solving an equation means finding the value(s) for the variable(s) that make the equation true. For linear equations, this typically involves manipulating the equation using algebraic operations to isolate one or more variables.
In the substitution method, the process begins with solving one of the equations for one variable in terms of another. This involves simple algebraic steps like:

• Adding or subtracting terms from both sides of the equation.
• Multiplying or dividing both sides of the equation by a number.

In our example, the first step was to solve the equation \( x + 2y = -1 \) for \( x \), giving us \( x = -1 - 2y \). This expression for \( x \) allows us to substitute into the other equation, aiding in solving for the other variable.

The substitution technique is a powerful tool for solving systems of linear equations. It involves substituting an expression derived from one equation into another. Here’s the step-by-step breakdown:
First, choose one of the equations and solve it for one variable. This can be the easier equation out of the two. In our problem, we opted for \( x = -1 - 2y \) from the first equation.
Next, substitute the expression for this variable into the other equation. Replace \( x \) in \( 2x - 3y = 12 \) with \( -1 - 2y \), resulting in a single equation in terms of \( y \).
Now, solve this new equation for \( y \). This involved simplifying \( 2(-1 - 2y) - 3y = 12 \) to \( y = -2 \).
Finally, use the value of \( y \) to find \( x \) by substituting \( y = -2 \) back into the earlier expression \( x = -1 - 2y \). This gives us \( x = 3 \).
Verification is key: Plugging these values back into the original equations confirmed they are correct. Both equations are satisfied with \( x = 3 \) and \( y = -2 \), showing the efficiency and accuracy of the substitution technique.