Solving Equations with Addition

In order to solve an equation, you must determine what the variable would have to equal in order for the equation to be true.

The goal in solving equations is to get the variable all by itself on one side of the equal sign. In order to do that, you need to move everything else to the other side of the equation.

While you are working with an equation, you have to keep it balanced. What that means is that you can add any number to an equation as long as you do it to both sides. Imagine a perfectly balanced hanging scale. If you add 5 oz. to one side you'd have to add 5 oz. to the other side in order to keep it balanced.

If a number is added to the variable, we will move it by adding its opposite to both sides of the equation. (This goes for subtraction as well because subtraction is really just adding a negative number.)

Example:

1) x-1=-3 Even if you can do this in your head, you need to learn the
steps so that you can apply them to more complex problems. Our goal is to get the x by itself, so we first need to figure out what needs to move. The -1 is on the same side as the x, so that is what we need to move. It's being added (remember subtraction is just adding a negative), so we are going to add the opposite to both sides. The opposite of -1 is +1, so
x-1+1=-3+1we will add +1 to both sides:
Now we will work with each side. On the left hand side,
x+0=-2 -1+1=0 and on the right, -3+1=-2. Then
x=-2x+0=x
We have the variable all by itself, so this is our answer.
x-1=-3We can check the answer by putting -2 back in the
-2-1=-3true original equation in place of x.

Final answer: x=-2 This is true, so we know we have the right answer.


2) x+8=-5 Our goal is to get the x by itself, so we first need to figure
out what needs to move. The 8 is on the same side as the x, so that is what we need to move. It's being added, so we are going to add the opposite to both sides. The opposite of 8 is -8, so we will add -8 to
x+8+(-8)=-5+(-8)both sides:
Now we will work with each side. On the left hand side,
x=-13 8+(-8)=0 so that will go away, and on the right,
-5+(-8)=-13.
We have the variable all by itself, so this is our answer.
x+8=-5 We can check the answer by putting -13 back in the
-13+8=-5true original equation in place of x.

Final answer: x=-13 This is true, so we know we have the right answer.


3) 7=y+10 If it confuses you to have the variable on the right hand
y+10=7 side, you can simply switch the sides of the equation.
We need to figure out what needs to move. The 10 is on the same side as the y, so that is what we need to move. It's being added, so we are going to add the opposite to both sides. The opposite of 10 is -10, so
y+10+(-10)=7+(-10) we will add -10 to both sides:
On the left hand side, 10+(-10)=0 so that will go
y=-3 away, and on the right, 7+(-10)=-3.
We have the variable all by itself, so this is our answer.
7=y+10 We can check the answer by putting -3 back in the
7=-3+10true original equation in place of y.

Final answer: y=-3 This is true, so we know we have the right answer.


Practice:Solve the equation and check your answer.

1) y-8=-2

2) x+3=-1

3) a+7=3

4) -6=x+1

5) 9+y=4

Answers: 1 y=62) x=-43) a=-44) x=-75) y=-5

Related Links:
Math
Algebra
Equations
Algebra Topics


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