Solving System Of Equations: X + Y = 7, Xy = 12

October 10, 2025 by TextBrain Team 48 views

Hey guys! Today, we're going to dive into solving a system of equations that looks pretty neat. Specifically, we'll tackle the system:

 x + y = 7
 xy = 12

This is a classic algebra problem, and I'm stoked to walk you through it step by step. So, grab your favorite beverage, maybe a snack, and let's get started!

Understanding the Problem

Before we jump right into solving, let's make sure we understand what's going on. A system of equations is simply a set of two or more equations that we're trying to solve simultaneously. In other words, we want to find values for x and y that satisfy both equations at the same time. In our case, we have two equations:

A linear equation: x + y = 7
A non-linear equation: xy = 12

Our goal is to find the pair(s) of x and y that make both of these equations true. There are a few ways we could approach this, but we're going to use the substitution method. This method involves solving one equation for one variable and then substituting that expression into the other equation. Trust me, it's easier than it sounds!

Step-by-Step Solution

Step 1: Solve for One Variable

Let's start with the first equation: x + y = 7. We can easily solve for either x or y. I'm going to solve for x, but you could just as easily solve for y – you'll get the same answer in the end. To solve for x, we simply subtract y from both sides of the equation:

 x = 7 - y

Now we have an expression for x in terms of y. This is key to our next step.

Step 2: Substitute into the Other Equation

Next, we're going to substitute our expression for x into the second equation: xy = 12. Wherever we see an x in the second equation, we're going to replace it with (7 - y). This gives us:

 (7 - y)y = 12

Now we have a single equation with only one variable, y. This is something we can solve! Let's simplify and rearrange the equation.

Step 3: Simplify and Rearrange

First, distribute the y:

 7y - y^2 = 12

Now, let's rearrange the equation into a standard quadratic form (i.e., ay^2 + by + c = 0). To do this, we'll move all the terms to one side of the equation:

 0 = y^2 - 7y + 12

Or, equivalently:

 y^2 - 7y + 12 = 0

Step 4: Solve the Quadratic Equation

We now have a quadratic equation in the form y^2 - 7y + 12 = 0. There are several ways to solve quadratic equations, including factoring, completing the square, or using the quadratic formula. In this case, the equation is easily factorable. We're looking for two numbers that multiply to 12 and add up to -7. Those numbers are -3 and -4. So, we can factor the quadratic equation as:

 (y - 3)(y - 4) = 0

To find the solutions for y, we set each factor equal to zero:

 y - 3 = 0  or  y - 4 = 0

Solving for y in each case gives us:

 y = 3  or  y = 4

So, we have two possible values for y: 3 and 4.

Step 5: Find the Corresponding Values of x

Now that we have the values for y, we need to find the corresponding values for x. We can use the expression we found earlier: x = 7 - y. Let's plug in each value of y:

If y = 3:
```
 x = 7 - 3 = 4
```
If y = 4:
```
 x = 7 - 4 = 3
```

So, we have two pairs of solutions: (4, 3) and (3, 4).

The Solutions

Therefore, the solutions to the system of equations are:

 x = 4, y = 3

and

 x = 3, y = 4

These are the two pairs of values for x and y that satisfy both equations in the system. You can check these solutions by plugging them back into the original equations to make sure they work.

Verification

Let's verify our solutions by substituting them back into the original equations:

For the solution x = 4, y = 3:

Equation 1: x + y = 7
```
 4 + 3 = 7  (True)
```
Equation 2: xy = 12
```
 4 * 3 = 12  (True)
```

For the solution x = 3, y = 4:

Equation 1: x + y = 7
```
 3 + 4 = 7  (True)
```
Equation 2: xy = 12
```
 3 * 4 = 12  (True)
```

Both solutions satisfy both equations, so we know we did it right!

Alternative Methods

While we used the substitution method to solve this system of equations, there are other methods you could use as well. Here are a couple of alternatives:

Elimination Method

The elimination method involves manipulating the equations so that when you add or subtract them, one of the variables is eliminated. In this case, it's a bit trickier to apply the elimination method directly because of the xy term in the second equation. However, you could still use it in conjunction with other techniques.

Graphical Method

You could also solve this system of equations graphically. To do this, you would graph both equations on the same coordinate plane and find the points where the graphs intersect. The coordinates of those points would be the solutions to the system. In this case, the graph of x + y = 7 is a straight line, and the graph of xy = 12 is a hyperbola. The points of intersection would be (4, 3) and (3, 4).

Common Mistakes to Avoid

When solving systems of equations, there are a few common mistakes that people often make. Here are some things to watch out for:

Incorrect Substitution: Make sure you substitute the expression for the variable correctly. It's easy to make a mistake when you're dealing with parentheses or negative signs.
Algebra Errors: Be careful when simplifying and rearranging equations. A small algebra error can throw off your entire solution.
Forgetting to Find Both Variables: Remember that you need to find values for both x and y to solve the system of equations. Don't stop after you find one variable; make sure to plug it back in to find the other.
Not Checking Your Solutions: Always check your solutions by plugging them back into the original equations to make sure they work. This can help you catch any mistakes you may have made along the way.

Conclusion

And that's it! We've successfully solved the system of equations:

 x + y = 7
 xy = 12

We found that the solutions are x = 4, y = 3 and x = 3, y = 4. I hope this step-by-step guide has been helpful. Remember, practice makes perfect, so keep solving those equations!

If you have any questions or want to try more examples, feel free to ask. Happy solving, folks! 🥳