Logarithmic Equations

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A second-order equation system is a set of two equations that each include a variable that is raised to the second power, like x^2 or y^2. These equations can be written in the form of:

ax^2 + by^2 = c dx + ey = f

To solve this kind of system, you'll need to use a method called substitution. Here's an example of how it works:

Let's say we have the following system of equations:

2x^2 + y^2 = 13 x + y = 5

First, we can use the second equation to solve for one of the variables in terms of the other. In this case, we can solve for x:

x = 5 - y

Next, we can substitute that expression for x into the first equation:

2(5-y)^2 + y^2 = 13

Now, we just need to solve for y. We can simplify the left side of the equation by expanding the square:

2(25 - 10y + y^2) + y^2 = 13 50 - 20y + 3y^2 = 13

This simplifies to:

3y^2 - 20y + 37 = 0

We can solve for y using the quadratic formula:

y = (20 ± √(-20^2 - 4337)) / (2*3)

After doing the math, we get two solutions for y:

y = 2 or y = (5/3)

Now we can plug each value of y back into the equation we used to solve for x:

If y = 2, then x = 3 If y = (5/3), then x = (10/3)

So the solutions to this system are:

x = 3, y = 2 x = (10/3), y = (5/3)

That's how you can solve a second-order equation system using substitution.

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