cgarcia71 said:
It seems as I am really having difficulties understanding how to solve equations. I need help with the following.
I need to solve each system by the substitution method and then determine whether or not the equations are independent, dependent or inconsistent.
x=y+3
3x-2y=4
I am not sure if I am supposed to do anything with the 1st equation because it doesn't seem to be in the right format.
am I supposed to take the second equation and plug do this
3x - 2 (y+3) = 4
Please help me understand... Thanks!
Hi! You can "plug in" a lamp....but in mathematics, we "substitute" a value or an expression for something that it is equal to.
The first equation,
x = y + 3
tells you that (y + 3) is the same thing as x, and can be used as a replacement for x.
The second equation,
3x - 2y = 4
has an "x" in it. So, in place of that "x", you can substitute something that you know is equal to x, namely (y + 3).
Let's do that substitution:
3x - 2y = 4
3(y+ 3) - 2y = 4
Now, we have an equation with just
one variable, y. So, we'll solve it. Start by multiplying each term inside the parentheses by the 3 which is outside:
3y + 3(3) - 2y = 4
3y + 9 - 2y = 4
Next, combine like terms to simplify the expression on the left side of the equals sign. 3y - 2y is 1y, or just y:
y + 9 = 4
Now, we want the variable term to be by itself on the left side. To "undo" the addition of 9,
subtract 9 .....but what you do to one side of the equation, you must do to the other side of the equation. So, we'll subtract 9 from sides of the equation:
y + 9 - 9 = 4 - 9
y = -5
So, we have a value for y. We also need a value for x. The first equation tells us that
x = y + 3
We know that y = -5. Substitute -5 for y in
x = y + 3
x = -5 + 3
x = -2
Ok....we have a value for x, and a value for y. So, we have a solution for the system. But, we had better check. Are
both equations in the system true when x = -2 and y = -5?
First equation:
x = y + 3
-2 = -5 + 3
-2 = -2 True
Second equation:
3x - 2y = 4
3(-2) - 2(-5) = 4
-6 + 10 = 4
4 = 4 True.
It checks. So, the solution for the system is x = -2 and y = -5.
I hope this helps you.
