91Ó°ÊÓ

Systems of equations involve finding values of variables that satisfy multiple equations simultaneously. This means that we have more than one equation working together to define the relationships between the variables. For instance, in the given exercise, we have two equations:
1. \(3m + n = 7\)
2. \(m + 2n = 9\)
Each equation provides different information about the variables 'm' and 'n'. Our goal is to find values for 'm' and 'n' that make both equations true at the same time. This is known as 'solving the system'. By solving systems of equations, we can find points where the lines represented by these equations intersect. This method is essential in various fields such as mathematics, physics, engineering, and economics.

Solving by elimination is one of the most effective methods to solve systems of linear equations. The basic idea behind this method is to remove one variable by adding or subtracting the equations, making it easier to solve for the remaining variable. Here’s a brief step-by-step outline for elimination:

• First, align and rewrite the equations if necessary.
• Next, adjust the coefficients to make them equal for the variable you plan to eliminate.
• Add or subtract the equations to eliminate that variable.
• Solve for the remaining variable.
• Substitute the value back into one of the original equations to find the other variable.

In the exercise provided, we adjusted the coefficients of ‘m’ so we could subtract the equations to eliminate ‘m’ and solve for ‘n’. This required multiplying Equation [B] by 3 to align the coefficients:
\[3(m + 2n) = 3(9)\]
Which gave us:
\[3m + 6n = 27\]
We then subtracted this new equation from Equation [A] to eliminate ‘m’.

Algebraic manipulation involves rearranging and simplifying equations to isolate variables and solve problems. This can include adding, subtracting, multiplying, or dividing both sides of an equation by the same number, combining like terms, and using properties of equality. Proper manipulation is critical when using the elimination method.
In our example, we performed several algebraic manipulations:

• We multiplied Equation [B] by 3 to match the coefficient of ‘m’ in Equation [A].
• We then subtracted the new equation from Equation [A] to eliminate ‘m’.
• After eliminating ‘m’, we were left with an equation in ‘n’.
• We solved for ‘n’ by isolating it through division.

These steps illustrate the importance of algebraic manipulation in simplifying and solving equations. By carefully following these steps, we ensure that our solutions are correct and all variables are properly accounted for.

The substitution method is another common technique for solving systems of equations. Instead of focusing on elimination, this method involves solving one of the equations for one variable in terms of the other. We then substitute this expression into the other equation. This method is often used when one of the equations is already solved for one variable, or can be rearranged easily.
Although we primarily solved our exercise using elimination, we incorporated substitution after finding one of the variables (n = 4). Here’s how substitution was used:
After solving for ‘n’, we substituted \(n = 4\) back into one of the original equations to find ‘m’. Specifically, into Equation [B], we substituted ‘4’ for ‘n’:
\[m + 2(4) = 9\]
This simplified to:
\[m + 8 = 9\]
Then, we solved for ‘m’ by isolating it:
\[m = 9 - 8\]
\[m = 1\]
As these steps demonstrate, the substitution method can be used in conjunction with elimination to efficiently solve systems of equations, ensuring all variables are effectively addressed.