91Ó°ÊÓ

Solving equations, especially systems of linear equations, forms a fundamental part of algebra. It involves finding values for variables that satisfy all given equations. The substitution method is particularly useful when one of the equations is already solved for a particular variable.
To begin the process, identify a variable that can be easily substituted into another equation. For example, if you have equations like:

• \( 2x + 3y = 7 \)
• \( x = 2 \)

you can substitute the value of \( x \) from the second equation into the first.

This makes solving the system easier because you reduce two variables to one in a single equation. Once the substitution is made, solve for the remaining variable to find your solution.

Linear equations are equations of the first degree, meaning they have no exponents higher than one. A typical form of a linear equation in two variables is \( ax + by = c \).
These equations represent straight lines on a graph. When solving them using the substitution method, it simplifies the problem to a matter of basic arithmetic once a variable is substituted.

In the equation \( 2x + 3y = 7 \), by substituting \( x = 2 \), the equation becomes linear with just one variable: \( 4 + 3y = 7 \). Solving involves ordinary steps:

• Subtracting 4 from both sides, \( 3y = 3 \).
• Dividing both sides by 3, leading to \( y = 1 \).

Linear equations can be solved graphically, through substitution, or elimination strategies, each offering unique benefits depending on the particular set of equations.

After arriving at a solution, it is crucial to verify its correctness. Verification involves plugging the values back into the original equations to ensure they hold true.
In our previous example:

• With \( x = 2 \) substituted back into the equation \( x = 2 \), it obviously checks out as correct.
• For \( 2x + 3y = 7 \), substituting \( x = 2 \) and \( y = 1 \) gives: \( 2 \times 2 + 3 \times 1 = 7 \).
• The left-hand side equals the right-hand side, confirming the solution is accurate.

Verification boosts confidence in the solution and ensures no calculation errors were made during the process. It solidifies understanding and encourages double-checking work before finalizing answers.