91Ó°ÊÓ

Simultaneous equations, also known as systems of equations, are sets of equations with multiple variables. The main goal is to find a common solution for these variables that satisfies all equations in the system. Each equation provides a different constraint, and the solution is where these constraints intersect.

For instance, in the provided exercise, we have:
\[ x + 3y = 6 \]
\[ 5x + 3y = -6 \]
We want to find values of \(x\) and \(y\) that make both equations true at the same time. This can be visualized as finding the common point on the graph where the lines intersect.

One way to solve simultaneous equations is using algebraic methods. These include substitution, elimination, and matrix methods. Let's focus on the elimination method used in the given solution.

First, write down the equations:
\[ x + 3y = 6 \]
\[ 5x + 3y = -6 \]
Notice they both contain the term \(3y\). We can subtract one equation from the other to eliminate \(y\) and solve for \(x\).
Subtracting the first from the second:
\[ (5x + 3y) - (x + 3y) = -6 - 6 \]
This results in:
\[ 4x = -12 \]
Solving for \(x\):
\[ x = -3 \]
Then substitute \(x = -3\) back into one of the original equations to find \(y\):
\[ -3 + 3y = 6 \]
Solve for \(y\):
\[ 3y = 9 \]
\[ y = 3 \]
So, the solution is \(x = -3\) and \(y = 3\).

Graphing is another method to solve systems of equations, giving a visual representation of the solution. For the given system:

1. Start by rewriting each equation in slope-intercept form, i.e., \(y = mx + b\).
For \(x + 3y = 6\), solve for \(y\):
\[ 3y = -x + 6 \]
\[ y = -\frac{1}{3}x + 2 \]
For \(5x + 3y = -6\), solve for \(y\):
\[ 3y = -5x - 6 \]
\[ y = -\frac{5}{3}x - 2 \]
2. Plot the lines on a graph.
3. The point where the lines intersect is the solution.
In our case, the lines intersect at \((-3, 3)\), confirming the solution found algebraically.

The substitution method involves solving one equation for one variable and then substituting this into the other equation. This method is particularly useful when one equation is easily solvable for one variable. Here’s how you can use this method for our system:

1. Solve the first equation for \(x\):
\[ x + 3y = 6 \]
\[ x = 6 - 3y \]
2. Substitute this expression for \(x\) into the second equation:
\[ 5x + 3y = -6 \]
\[ 5(6 - 3y) + 3y = -6 \]
Simplify:
\[ 30 - 15y + 3y = -6 \]
\[ 30 - 12y = -6 \]
Solve for \(y\):
\[ -12y = -36 \]
\[ y = 3 \]
3. Substitute \(y = 3\) back into the expression for \(x\):
\[ x = 6 - 3(3) \]
\[ x = -3 \]
Thus, the solution \((x, y)\) is \((-3, 3)\).