Lesson Objectives
- Demonstrate an understanding of the definition of a Term
- Demonstrate an understanding of the definition of an Algebraic Expression
- Learn how to tell the difference between an Algebraic Expression and an Equation
- Learn how to determine if a given value represents a solution for an Equation
Algebraic Expressions vs. Equations
In our last lesson, we learned the definition of an algebraic expression. Recall that an algebraic expression is a single term or a collection of terms separated by "+" or "-" signs: 
In our image above, we see an example of an algebraic expression.
8x - 1 has two terms (8x) and (1), which are separated by the "-" sign. Let's look at another example of an algebraic expression:
5x2 + 6x - 7 has three terms (5x2), (6x), and (7), which are separated by a "+" sign and a "-" sign.
Notice how algebraic expressions do not contain the "=" equality sign. Algebraic expressions can only be simplified by combining like terms or evaluated if values are given for the variables. Let's take a look at a few examples.
Example 1: Simplify each by combining like terms.
7x2 - 2x2 + 3x - 5x - 1
To simplify here, we combine like terms. Remember, like terms have the same variable(s) raised to the same power(s).
Like Terms » 7x2, 2x2
Like Terms » 3x, 5x
When we combine like terms, we keep the variable part the same, and perform the operations with the coefficients:
7x2 - 2x2 = (7 - 2)x2 = 5x2
3x - 5x = (3 - 5)x = -2x
Putting everything together gives us:
5x2 - 2x - 1
7x2 - 2x2 + 3x - 5x - 1 = 5x2 - 2x - 1
Example 2: Evaluate -3x2 - x + 7 for x = 2 and x = -1
To evaluate an algebraic expression for a given value, we plug the value in for each occurrence of the variable.
We will begin by evaluating for x = 2, this means we will plug in a 2 for each x in our expression:
-3x2 - x + 7
-3(2)2 - 2 + 7
-3(2)2 - 2 + 7 = -3(4) - 2 + 7
-3(4) - 2 + 7 = -12 - 2 + 7
-12 - 2 + 7 = -14 + 7 = -7
Evaluating -3x2 - x + 7 for x = 2, gives us a result of (-7).
Let's look at our second case, where x = -1. Here we plug in a -1 for each occurrence of x:
-3x2 - x + 7
-3(-1)2 - (-1) + 7
-3(-1)2 - (-1) + 7 = -3(1) + 1 + 7
-3(1) + 1 + 7 = -3 + 1 + 7
-3 + 1 + 7 = -2 + 7 = 5
Evaluating -3x2 - x + 7 for x = -1, gives us a result of (5).
In our above example, we can see that the algebraic expression (8x - 1) is set equal to (7x - 4). Notice the equality symbol from the start. When we have an equation, we can solve for the value that makes the equation "true". For an equation to be true, the left and right sides must be the same or equal in value. In our next lesson, we will start showing how to solve an equation using a step-by-step process. For now, let's focus on how to determine if a given value represents a solution.
Suppose we had the following equation:
x + 2 = 4
This equation is stating that some unknown value (x) plus 2 is equal to or is the same as 4. We can change this into a question:
? + 2 = 4
In other words, what value when added to 2 gives us 4? The answer is 2 since 2 + 2 = 4. In this case, x = 2.
The way we check this solution is by plugging in a 2 for x and evaluating:
x + 2 = 4
2 + 2 = 4
4 = 4
Once we have plugged in 2 for x and evaluated the left side, we see that we have the same value (4) on the left and right. This means our solution (x = 2) is correct. This is the great thing about solving equations. We can always check our answer and see if we got the correct solution. Let's take a look at a few examples.
Example 3: Determine if x = 5 is a solution to the equation 2x - 1 = 9
To solve this problem, we plug in a 5 for x and evaluate. We are looking for the left and right sides to be the same value.
2x - 1 = 9
2(5) - 1 = 9
2(5) - 1 = 9
10 - 1 = 9
9 = 9
Since we have the same value (9) on both sides of the equation, our solution (x = 5) is correct.
Example 4: Determine if x = -3 is a solution to the equation 2(x - 4) = 14
To solve this problem, we plug in a (-3) for x and evaluate. We are looking for the left and right sides to be the same value.
2(x - 4) = 14
2(-3 - 4) = 14
2(-3 - 4) = 14
2(-7) = 14
-14 = 14 (false!)
Since we do not have the same value on each side of the equation (-14 vs. +14), (x = -3) is not a solution to our equation.
Example 5: Determine if x = 11 is a solution to the equation -4x = -44
To solve this problem, we plug in an 11 for x and evaluate. We are looking for the left and right sides to be the same value.
-4x = -44
-4(11) = -44
-44 = -44
Since we have the same value (-44) on both sides of the equation, our solution (x = 11) is correct.
In our image above, we see an example of an algebraic expression. 8x - 1 has two terms (8x) and (1), which are separated by the "-" sign. Let's look at another example of an algebraic expression:
5x2 + 6x - 7 has three terms (5x2), (6x), and (7), which are separated by a "+" sign and a "-" sign. Notice how algebraic expressions do not contain the "=" equality sign. Algebraic expressions can only be simplified by combining like terms or evaluated if values are given for the variables. Let's take a look at a few examples.
Example 1: Simplify each by combining like terms.
7x2 - 2x2 + 3x - 5x - 1
To simplify here, we combine like terms. Remember, like terms have the same variable(s) raised to the same power(s).
Like Terms » 7x2, 2x2
Like Terms » 3x, 5x
When we combine like terms, we keep the variable part the same, and perform the operations with the coefficients:
7x2 - 2x2 = (7 - 2)x2 = 5x2
3x - 5x = (3 - 5)x = -2x
Putting everything together gives us:
5x2 - 2x - 1
7x2 - 2x2 + 3x - 5x - 1 = 5x2 - 2x - 1
Example 2: Evaluate -3x2 - x + 7 for x = 2 and x = -1
To evaluate an algebraic expression for a given value, we plug the value in for each occurrence of the variable.
We will begin by evaluating for x = 2, this means we will plug in a 2 for each x in our expression:
-3x2 - x + 7
-3(2)2 - 2 + 7
-3(2)2 - 2 + 7 = -3(4) - 2 + 7
-3(4) - 2 + 7 = -12 - 2 + 7
-12 - 2 + 7 = -14 + 7 = -7
Evaluating -3x2 - x + 7 for x = 2, gives us a result of (-7).
Let's look at our second case, where x = -1. Here we plug in a -1 for each occurrence of x:
-3x2 - x + 7
-3(-1)2 - (-1) + 7
-3(-1)2 - (-1) + 7 = -3(1) + 1 + 7
-3(1) + 1 + 7 = -3 + 1 + 7
-3 + 1 + 7 = -2 + 7 = 5
Evaluating -3x2 - x + 7 for x = -1, gives us a result of (5).
What is an Equation?
What is an equation? An equation is a statement that two algebraic expressions are equal. An equation always includes the equality symbol "=". In our above example, we can see that the algebraic expression (8x - 1) is set equal to (7x - 4). Notice the equality symbol from the start. When we have an equation, we can solve for the value that makes the equation "true". For an equation to be true, the left and right sides must be the same or equal in value. In our next lesson, we will start showing how to solve an equation using a step-by-step process. For now, let's focus on how to determine if a given value represents a solution. Suppose we had the following equation:
x + 2 = 4
This equation is stating that some unknown value (x) plus 2 is equal to or is the same as 4. We can change this into a question:
? + 2 = 4
In other words, what value when added to 2 gives us 4? The answer is 2 since 2 + 2 = 4. In this case, x = 2.
The way we check this solution is by plugging in a 2 for x and evaluating:
x + 2 = 4
2 + 2 = 4
4 = 4
Once we have plugged in 2 for x and evaluated the left side, we see that we have the same value (4) on the left and right. This means our solution (x = 2) is correct. This is the great thing about solving equations. We can always check our answer and see if we got the correct solution. Let's take a look at a few examples.
Example 3: Determine if x = 5 is a solution to the equation 2x - 1 = 9
To solve this problem, we plug in a 5 for x and evaluate. We are looking for the left and right sides to be the same value.
2x - 1 = 9
2(5) - 1 = 9
2(5) - 1 = 9
10 - 1 = 9
9 = 9
Since we have the same value (9) on both sides of the equation, our solution (x = 5) is correct.
Example 4: Determine if x = -3 is a solution to the equation 2(x - 4) = 14
To solve this problem, we plug in a (-3) for x and evaluate. We are looking for the left and right sides to be the same value.
2(x - 4) = 14
2(-3 - 4) = 14
2(-3 - 4) = 14
2(-7) = 14
-14 = 14 (false!)
Since we do not have the same value on each side of the equation (-14 vs. +14), (x = -3) is not a solution to our equation.
Example 5: Determine if x = 11 is a solution to the equation -4x = -44
To solve this problem, we plug in an 11 for x and evaluate. We are looking for the left and right sides to be the same value.
-4x = -44
-4(11) = -44
-44 = -44
Since we have the same value (-44) on both sides of the equation, our solution (x = 11) is correct.
Skills Check:
Example #1
Which answer is a solution to the given equation. $$-3(4 + 4x) + 2=-58$$
Please choose the best answer.
A
$$x=-3$$
B
$$x=6$$
C
$$x=-4$$
D
$$x=1$$
E
$$x=4$$
Example #2
Which answer is a solution to the given equation. $$4(3x - 2)=-44$$
Please choose the best answer.
A
$$x=2$$
B
$$x=-2$$
C
$$x=3$$
D
$$x=-3$$
E
$$x=5$$
Example #3
Which answer is a solution to the given equation. $$-x - 4(1 + 4x)=-72$$
Please choose the best answer.
A
$$x=-1$$
B
$$x=2$$
C
$$x=4$$
D
$$x=-4$$
E
$$x=-2$$
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