What is the process of finding the solution of an equation?

Answer 1

A system of equations simply means, what number can you plug in for X and Y that will make both equations work?

For example,

Solve the following system of equations:

x + y = 11

3x - y = 5

In order to solve a system of equations, you have 2 ways.

Method 1: Substitution

Choose 1 equation. It doesn't matter which!

To make it easier, pick an equation without alot of numbers multiplied to X or Y.

ie: x + y = 11.

Now, solve for X or Y. You pick! OK, I'll pick. (but it doesn't matter which. just pick either!)

I'll solve for X.

x + y = 11

Subtract y from both sides, because I want my equation to look like x = something.

x = 11 - y

Now that we've solved for a variable (x), plug what x is equal to (11-y) into the equation you didn't solve, wherever you see x. Make sure to use parenthesis!

3x - 5 = y

3(11-y) - 5 = y

Now use your algebra to simplify the equation.

33 - 3y -5 = y (distributive property -- this is why we use parenthesis....)

28 - 3y = y

28 = 4y

7 = y

Now that we have a number for Y, plug 7 in wherever you see Y into one of your original equations. We're trying to solve for X now. In fact, to make it easier, you can plug y = 7 into the equation we solved x for!

Remember, x = 11-y ?

x = 11 - (7)

x = 4

So we know y = 7 and x = 4. These numbers should work in BOTH original equations!! If they don't, you made a mistake and you should double check your algebra.

x + y = 11

(4) + (7) = 11

11 = 11 (check)

3x - y = 5

3(4) - (7) = 5

12 - 7 = 5

5 = 5 (check)

Method 2: Elimination

As this method suggests, we need to "eliminate" or get rid of 1 variable.

Line both equations up so the x's and y's are stacked on top of each other, and the = signs as well. Like you were going to do an addition, or subtraction problem.

x + y = 11

3x - y = 5

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What our goal is, is to eliminate one variable. Just like in method 1, you choose which variable. The easiest way is to pick a variable with no numbers in front.

Those y's look pretty good!

When using the elimination method we want 2 things to happen.

1) The number in front of the variable being eliminated is the same in both equations

2) The variable being eliminated must have opposite signs in both equations.

Luckily, our y variable has no coefficient (number in front) AND already has an opposite sign! (+y on top, -y on bottom). All we need to do now is add downward, like a regular addition problem.

1x + 1y = 11

3x - 1y = 5

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4x = 16

Notice the y's cancelled each other out? They "eliminated" themselves, which was our goal! Remember to add the numbers on the other side of the equals sign as well.

Now just solve for X.

4x = 16 -> x = 4

Once we find a value for x, substitute x=4 into either original equations, to solve for y.

x + y = 11

(4) + y = 11

y = 11 - 4

y = 7

So x =4 and y=7, just like in the first method. Use whichever makes you feel comfortable. I find substitution is much easier, but you might prefer elimination. Good luck!

Answer 2

It is known as solving the equation.