STANDARD 4 & 5:
SIMPLIFYING EXPRESSIONS & SOLVING EQUATIONS
Combining Like Terms
What are Like Terms?
The key to using and understanding the method of Combining Like Terms is to understand like terms and be able to identify when a pair of terms is a pair of like terms. Some examples of like terms are presented below.
The following are like terms because each term consists of a single variable, x, and a numeric coefficient.
2x, 45x, x, 0x, -26x, -x
For comparison, below are a few examples of unlike terms:
The following two terms both have a single variable with an exponent of 1, but the terms are not alike since different variables are used.
17x, 17z
Example 1:
Simplify 2x + 3y - 2 + 3x + 6y + 7
This expression can be simplified by identifying like terms and then grouping and combining like terms, like this:
· +2x and +3x are like terms, and can be combined to give +5x,
· +3y and +6y combine to give +9y, and
· -2 and +7 combine to give +5.
So after simplifying, this expression becomes: 5x + 9y + 5
The key to using and understanding the method of Combining Like Terms is to understand like terms and be able to identify when a pair of terms is a pair of like terms. Some examples of like terms are presented below.
The following are like terms because each term consists of a single variable, x, and a numeric coefficient.
2x, 45x, x, 0x, -26x, -x
For comparison, below are a few examples of unlike terms:
The following two terms both have a single variable with an exponent of 1, but the terms are not alike since different variables are used.
17x, 17z
Example 1:
Simplify 2x + 3y - 2 + 3x + 6y + 7
This expression can be simplified by identifying like terms and then grouping and combining like terms, like this:
· +2x and +3x are like terms, and can be combined to give +5x,
· +3y and +6y combine to give +9y, and
· -2 and +7 combine to give +5.
So after simplifying, this expression becomes: 5x + 9y + 5
Distributive Property
Simplify: 3b - (4b - 6b + 2) + b
The brackets in this expression can be removed first, then the expression may be simplified like example 1 above, OR the like terms: +4b and -6b inside the brackets can be combined to give -2b before removing the brackets.
·
Simplifying the bracketed expression we have: 3b - (-2b + 2) + b
There is a minus sign before the left bracket, so we must use the distributive property and distribute the negative one:
· Removing the brackets: 3b + 2b - 2 + b
· Grouping and combining the like terms: +3b, +2b and +b combine to give 6b:
6b - 2
Standard 4 Practice | |
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Solving Equations:
Multi-step and Word Problems
The whole point of "solving equations" is to get the variable alone on one side of an equation and a number on the other. In order to do this, we use inverse operations.
Inverse, means opposite. For example, the inverse of addition is subtraction. The inverse of multiplication is division. And for more complicated problems, the inverse of squaring something is taking the square root of it.
This brings up another main point, whatever you do to one side of the equation, you must also do to the other side. That is why we call these the "properties of equality."
Addition Property of equality: Adding a number to both sides of the equation.
Subtraction Property of Equality: Subtracting a number to both sides of the equation.
Multiplication Property of Equality: Multiplying a number on both sides of the equation.
Division Property of Equality: Dividing a number on both sides of the equation.
Example 1: Solve x + 6 = –3
• I want to get the x by itself; that is, I want to get "x" on one side of the "equals" sign, and some number on the other side.
• Since I want just x on the one side, this means that I don't like the "plus six" that's currently on the same side as the x.
• Since the 6 is added to the x, I need to subtract to get rid of it. That is, I will need to subtract a 6 from the x in order to "undo" having added a 6 to it. This is called the subtraction property of equality.
• This brings up the most important consideration with equations:
Whatever you do to an equation, do the SAME thing to B O T H sides of that equation!
x + 6 = -3
-6 -6
x = -9
Example 2: Solve 7x + 2 = –54
• I need to undo the "times seven" and the "plus two". There is no rule about which "undo" I should do first. However, if I first divide through by 7, I'm going to have fractions. Personally, I prefer to avoid fractions if possible, so I almost always do any adding/subtracting before any multiplying/dividing:
7x + 2 = –54
-2 -2 subtraction property of equality
7x = -56
7 7 division property of equality
x = 8
Example 3: Solve 4x - 6 = 6x
This problem can through students off because there is an x on both sides of the equation. Instead of freaking out, just relax and bring the x's together using a property of equality. In this case, to begin I will move the 4x to the right side of the equation first using the subtraction property of equality. Note: it does not matter which x you move, but it might be easier to move the smaller one (4 instead of 6) so that you can avoid negatives!
4x - 6 = 6x
-4x -4x subtraction prop of =
-6 = 2x
2 2 division prop of =
-3 = x
Inverse, means opposite. For example, the inverse of addition is subtraction. The inverse of multiplication is division. And for more complicated problems, the inverse of squaring something is taking the square root of it.
This brings up another main point, whatever you do to one side of the equation, you must also do to the other side. That is why we call these the "properties of equality."
Addition Property of equality: Adding a number to both sides of the equation.
Subtraction Property of Equality: Subtracting a number to both sides of the equation.
Multiplication Property of Equality: Multiplying a number on both sides of the equation.
Division Property of Equality: Dividing a number on both sides of the equation.
Example 1: Solve x + 6 = –3
• I want to get the x by itself; that is, I want to get "x" on one side of the "equals" sign, and some number on the other side.
• Since I want just x on the one side, this means that I don't like the "plus six" that's currently on the same side as the x.
• Since the 6 is added to the x, I need to subtract to get rid of it. That is, I will need to subtract a 6 from the x in order to "undo" having added a 6 to it. This is called the subtraction property of equality.
• This brings up the most important consideration with equations:
Whatever you do to an equation, do the SAME thing to B O T H sides of that equation!
x + 6 = -3
-6 -6
x = -9
Example 2: Solve 7x + 2 = –54
• I need to undo the "times seven" and the "plus two". There is no rule about which "undo" I should do first. However, if I first divide through by 7, I'm going to have fractions. Personally, I prefer to avoid fractions if possible, so I almost always do any adding/subtracting before any multiplying/dividing:
7x + 2 = –54
-2 -2 subtraction property of equality
7x = -56
7 7 division property of equality
x = 8
Example 3: Solve 4x - 6 = 6x
This problem can through students off because there is an x on both sides of the equation. Instead of freaking out, just relax and bring the x's together using a property of equality. In this case, to begin I will move the 4x to the right side of the equation first using the subtraction property of equality. Note: it does not matter which x you move, but it might be easier to move the smaller one (4 instead of 6) so that you can avoid negatives!
4x - 6 = 6x
-4x -4x subtraction prop of =
-6 = 2x
2 2 division prop of =
-3 = x
Word Problems can be FUN to solve, if only you know the clue words (see above)
Check out some of these word problem examples!
Example 1- Consecutive Numbers: http://www.virtualnerd.com/algebra-1/linear-equations-solve/word-problem-consecutive-numbers.php
Example 2- age problem: http://www.virtualnerd.com/algebra-1/linear-equations-solve/word-problem-division-example.php
Example 3- Proportion problem: http://www.virtualnerd.com/algebra-1/linear-equations-solve/proportion-word-problem-set-up.php
Example 4- multistep: http://www.virtualnerd.com/algebra-2/equations-inequalities/word-problem-multi-step-equation-example.php
Example 1- Consecutive Numbers: http://www.virtualnerd.com/algebra-1/linear-equations-solve/word-problem-consecutive-numbers.php
Example 2- age problem: http://www.virtualnerd.com/algebra-1/linear-equations-solve/word-problem-division-example.php
Example 3- Proportion problem: http://www.virtualnerd.com/algebra-1/linear-equations-solve/proportion-word-problem-set-up.php
Example 4- multistep: http://www.virtualnerd.com/algebra-2/equations-inequalities/word-problem-multi-step-equation-example.php
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