Solving Simultaneous Equations: Substitution & Elimination
Hey guys! Today, we're diving into the fascinating world of simultaneous equations. Specifically, we're going to tackle the following system of equations:
9x - 9y = 27
x - y = 3
We'll be solving this using two popular methods: substitution and elimination. So, buckle up, grab your pencils, and let's get started!
a) Solving by Substitution Method
The substitution method involves solving one equation for one variable and then substituting that expression into the other equation. This effectively reduces the system of two equations into a single equation with one variable, which we can then easily solve. Let's break it down step-by-step:
Step 1: Simplify the Equations (Optional but Recommended)
Before we jump into the substitution, let's simplify the first equation. Notice that all the coefficients in the equation 9x - 9y = 27 are divisible by 9. Dividing the entire equation by 9, we get:
x - y = 3
Woah! Look at that! The first equation simplifies to the same as the second equation. This already gives us a hint that there might be something interesting about this system of equations. But let's proceed with the substitution method to see how it plays out.
Step 2: Solve One Equation for One Variable
Let's choose the second equation, x - y = 3, and solve it for x. This is a straightforward step. We simply add y to both sides of the equation:
x - y + y = 3 + y
x = 3 + y
Great! We now have x expressed in terms of y. This is our key to the substitution.
Step 3: Substitute the Expression into the Other Equation
Now, we take the expression we found for x, which is 3 + y, and substitute it into the other equation. Since we used the second equation to solve for x, we'll substitute into the (simplified) first equation, which is also x - y = 3:
(3 + y) - y = 3
See what we did there? We replaced the x in the equation with (3 + y). Now we have an equation with only one variable, y.
Step 4: Solve for the Remaining Variable
Let's simplify and solve the equation we obtained in the previous step:
3 + y - y = 3
3 = 3
Wait a minute... 3 = 3? This is a true statement, but it doesn't give us a specific value for y. This is a crucial observation! It means that the equation is always true, regardless of the value of y. This indicates that we have infinitely many solutions.
Step 5: Express the Solution in Terms of the Parameter
Since we have infinitely many solutions, we can express the solution in terms of a parameter. Let's let y = t, where t can be any real number. Now, we can substitute this back into the equation x = 3 + y to find x in terms of t:
x = 3 + t
Therefore, the solution to the system of equations using the substitution method is:
x = 3 + t
y = t
Where t is any real number. This means that for every value we choose for t, we get a valid solution (x, y) for the system of equations. For instance, if t = 0, then x = 3 and y = 0. If t = 1, then x = 4 and y = 1, and so on.
b) Solving by Elimination Method
The elimination method involves manipulating the equations in such a way that when we add or subtract them, one of the variables cancels out. This leaves us with a single equation with one variable, which we can solve. Let's walk through the process:
Step 1: Align the Variables
First, we need to make sure the variables are aligned in both equations. Our equations are already in a good format:
9x - 9y = 27
x - y = 3
The x terms are aligned, the y terms are aligned, and the constants are aligned.
Step 2: Multiply Equations to Create Opposing Coefficients
To eliminate a variable, we need to make the coefficients of either x or y opposites of each other. Let's choose to eliminate x. To do this, we can multiply the second equation by -9:
-9(x - y) = -9(3)
-9x + 9y = -27
Now, our system of equations looks like this:
9x - 9y = 27
-9x + 9y = -27
Notice that the coefficients of x are now opposites (9 and -9).
Step 3: Add the Equations
Now, we add the two equations together:
(9x - 9y) + (-9x + 9y) = 27 + (-27)
9x - 9y - 9x + 9y = 0
0 = 0
Again, we end up with 0 = 0, which is a true statement but doesn't give us specific values for x or y. This confirms our earlier finding that the system has infinitely many solutions.
Step 4: Express the Solution in Terms of a Parameter
As we saw with the substitution method, we need to express the solution in terms of a parameter. We can use the same approach. Let y = t, where t is any real number. We can substitute this into either of the original equations (or the simplified forms) to find x in terms of t. Let's use the second equation, x - y = 3:
x - t = 3
x = 3 + t
Thus, the solution to the system of equations using the elimination method is:
x = 3 + t
y = t
Where t is any real number. This is the same solution we obtained using the substitution method.
Conclusion: Infinite Solutions and What It Means
So, guys, we've successfully solved the given system of simultaneous equations using both the substitution and elimination methods. The key takeaway here is that we arrived at the same result using both methods: an infinite number of solutions expressed as x = 3 + t and y = t, where t is any real number.
This means that the two original equations represent the same line! If you were to graph these equations, you'd see that they overlap completely. This is why we don't get a unique solution (a single point where the lines intersect), but rather a whole set of solutions (all the points on the line).
Isn't math fascinating? Keep exploring, keep questioning, and keep solving! You've got this!