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Tips for Successfully Solving Equations
One of my beginning Algebra students asked me the following question about solving equations - “How do I know when to keep the signs or change them when solving equations?” I thought, “Good question. Others may want to know the answer to this same question as well.”
Familiarize yourself with the basic Algebra vocabulary relating to solving equations.
Variable - unknown value represented by a letter or symbol
Coefficient - the number in front of a variable that shows the amount by which the variable is multiplied
Constant - known value represented by a number and does not contain a variable
Term - a variable (with or without a coefficient) or a number (constant)
Like Terms - terms that are alike either because they contain the same variable or because they are constants
Combining Like Terms - collecting terms on the same side of the equation
In the next two equations, the first does not require the signs to be changed in order to solve while the second equation requires changing the signs.
x = -4 + 7
Since x is already isolated, meaning it is already alone on one side of the equation; there is no need to change any of the signs. Simply add 7 + (-4) to get 3.
x = 3
In the next equation, the signs must be changed to solve for y. In Algebra, this is called using additive inverses or opposites: two numbers whose sum is 0 such that a + (-a) = 0.
4y - 4 + y = 6y + 20 - 4y
Notice that the variable y is present on the left and the right sides of the equation. In order to solve for y, all of the y terms must be isolated on one side and all of the constant terms must be isolated on the other side of the equal sign.
Opposites are used to move terms to combine like terms. Anytime a term (variable or constant) is moved to the opposite side of the equation, the sign in front of the term is changed to its opposite. Think of it like this: Whenever a move is required, signs must be changed; just like when we move from one location to the next, our address is changed.
First, move all of the y’s on the right side of the equation to the left side by “changing their addresses.” Keep in mind that whatever change is made on one side of the equation, the exact same change must be made on the opposite side of the equation.
4y - 4 + y - 6y + 4y = 6y (- 6y) + 20 - 4y (+ 4y) Opposites will cancel each other.
4y - 4 + y - 6y + 4y = 20
Move -4 to the right side of the equation by changing its address or sign.
4y - 4 (+ 4) + y - 6y + 4y = 20 (+ 4)
4y + y - 6y + 4y = 20 (+ 4)
Now that all like terms have been combined, simplify the equation by adding/subtracting like terms. Pay close attention to and follow the signs in front of the numbers. They are important for knowing when to add or subtract.
4y + y - 6y + 4y = 20 (+ 4)
5y - 6y + 4y = 24
-y + 4y = 24
3y = 24
Continue to think opposites. 3y indicates multiplication. The opposite of multiplication is division. Therefore, the last step is to divide both sides by 3.
3y/3 = 24/3
y = 8 (24/3)
Make a habit of checking your answers. Teachers and instructors love when students do this. Additionally, it helps you to know whether or not your answer is correct. Therefore, you can fix problems before submitting your assignment, test, etc.
Check y = 8 by substituting (8) into the original equation.
4(8) - 4 + 8 = 6(8) + 20 - 4(8)
32 - 4 + 8 = 48 + 20 - 32
28 + 8 = 68 - 32
36 = 36
If the solution is true, then the answer is correct.