Simplify the expression. 1 2 x 2 + 3 4 x y − 1 3 x 2 + 1 4 x y \frac12x^2+\frac34xy-\frac13x^2+\frac14xy

5 6 x 2 − 1 2 x y \frac56x^2-\frac12xy

1. Review of Real Numbers

Simplifying Expressions

1. Review of Real Numbers

Simplifying Expressions: Videos & Practice Problems

Topic summary

Algebraic expressions consist of terms separated by plus or minus signs, where like terms share identical variables raised to the same exponents. Combining like terms involves adding or subtracting their numerical coefficients, simplifying expressions by eliminating parentheses and grouping like terms. This process is essential for working with polynomials, including multivariable polynomials, and understanding concepts like coefficients, exponents, and powers. Mastery of simplifying expressions enhances problem-solving skills and prepares students for advanced topics such as binomials, trinomials, and polynomial degrees, all foundational in algebra and polynomial manipulation.

concept

Like Terms

Video duration:

Like Terms Video Summary

Algebraic expressions consist of numbers and variables combined through operations such as addition and subtraction. The individual parts of an expression separated by plus or minus signs are called terms. For example, in the expression 2x + 5, the terms are 2x and 5.

Like terms are terms that contain the exact same variables raised to the same exponents. The order of variables does not affect whether terms are like terms because multiplication is commutative. For instance, 4x² and 7x² are like terms since both have the variable x raised to the power of 2. Similarly, 2ab and 8ba are like terms because they contain the same variables a and b, each raised to the first power (implied exponent 1). Another example is 5x²yz and -10x²yz, where the variables x, y, and z match exactly with the same exponents.

It is important to note that the coefficients (numerical factors) of terms do not determine whether terms are like terms. Only the variables and their exponents matter.

Unlike terms differ either in their variables or in the exponents of those variables. For example, 4x² and 7x³ are not like terms because the exponents of x differ (2 versus 3). Likewise, 2a and 2ba are not like terms because the second term contains an additional variable b. Another example is 5x²yz and -10xy²z, which are unlike terms since the exponents of x and y differ between the two terms.

Combining like terms is a fundamental skill in algebra. This process involves adding or subtracting the coefficients of like terms while keeping the variable part unchanged. For example, combining 4x² + 7x² results in 11x² by adding the coefficients 4 and 7. Similarly, 2ab + 8ba can be combined as 10ab after recognizing the terms as like terms and adding their coefficients.

It is crucial to avoid combining unlike terms, as their variable parts differ. For instance, 5x²yz - 10xy²z cannot be combined because the exponents of x and y differ, making them unlike terms.

Understanding how to identify terms, distinguish like terms from unlike terms, and correctly combine like terms lays the foundation for simplifying algebraic expressions and solving equations efficiently.

example

Like Terms Example 1

Video duration:

Like Terms Example 1 Video Summary

When combining like terms in algebraic expressions, it is essential to identify terms that have the same variable raised to the same exponent. For example, in the expression 6a² + 8a², both terms contain the variable a raised to the power of 2. Since these are like terms, their coefficients can be added together, resulting in 14a². This process involves adding the numerical coefficients while keeping the variable and its exponent unchanged.

In contrast, consider the expression 7x² + 8x + 4. Although the first two terms share the variable x, the exponents differ: the first term has x squared (x²), while the second term has x to the first power (implied exponent of 1). Because the exponents are not the same, these terms are not like terms and cannot be combined. The constant term 4 stands alone as it has no variable, so it also cannot be combined with the other terms.

Finally, in the expression -5y⁵ + 3y⁵, both terms have the variable y raised to the fifth power, making them like terms. Combining their coefficients, -5 + 3, results in -2y⁵. This demonstrates that when combining like terms, the coefficients are added algebraically, taking into account their signs, while the variable and exponent remain constant.

Understanding how to combine like terms is fundamental in simplifying algebraic expressions, which is a critical skill in solving equations and manipulating polynomials. Recognizing that like terms must have identical variables raised to the same powers ensures accurate simplification and lays the groundwork for more advanced algebraic operations.

example

Like Terms Example 2

Video duration:

Like Terms Example 2 Video Summary

When simplifying algebraic expressions, it is important to combine like terms so that each variable appears only once. Like terms are terms that have the same variable raised to the same exponent. For example, in the expression 3p - 9p + 4 - 2 + p, the terms 3p, -9p, and p are like terms because they all contain the variable p raised to the first power (an invisible exponent of 1). To combine these, you add or subtract their coefficients: 3 minus 9 equals -6, and then adding 1 gives -5. This results in -5p.

Next, consider the constants 4 and -2. Constants are like terms because they have no variable attached, which means their variable is effectively zero. You can combine these by simple addition or subtraction: 4 minus 2 equals 2.

After combining all like terms, the simplified expression is -5p + 2. This process of combining like terms helps to write expressions in their simplest form, making them easier to understand and work with in further algebraic operations.

example

Like Terms Example 3

Video duration:

Like Terms Example 3 Video Summary

When simplifying algebraic expressions, it is important to combine like terms so that each variable appears only once. Consider the expression 6z - 4 + 11z + 9. To identify like terms, look for terms that have the same variable raised to the same exponent. Here, both 6z and 11z contain the variable z with an implied exponent of 1, making them like terms. You can combine these by adding their coefficients: \$6 + 11 = 17$, resulting in $17z\(.

Next, focus on the constant terms, which are numbers without variables. In this expression, the constants are -4 and 9. Since they are like terms (both constants), you can add them directly: \)-4 + 9 = 5\(.

After combining like terms, the simplified expression is \)17z + 5$. This expression has the variable z appearing only once and a single constant term, making it fully simplified.

Problem

Combine like terms such that each variable only appears once.
$\frac{3}{4} x + \frac{1}{2} x$

$\frac{4}{6} x$

$\frac{3}{8} x$

$\frac{5}{4} x$

$\frac{1}{4} x$

Problem

Combine like terms such that each variable only appears once.
$0.5 z + .25 z - 1.35 + 1.55$

$0.75 z minus 0.2$

$0.75 z plus 0.2$

$0.95 z$

$negative 2.15 z$

concept

Simplify Expressions

Video duration:

Simplify Expressions Video Summary

To simplify algebraic expressions effectively, it is essential to understand that a fully simplified expression contains no parentheses and has all like terms combined. The process begins by distributing any constants or variables across parentheses. For example, when you have an expression like −z + 2(3 + 5z), distribute the 2 to both 3 and 5z, resulting in −z + 6 + 10z. This step ensures that all terms are expressed without parentheses, making it easier to identify like terms.

Like terms are terms that share the same variable raised to the same exponent. In the expression above, −z and 10z are like terms because both contain the variable z to the first power. Constants, such as 6, are not like terms with variables and remain separate. Grouping like terms together, such as writing −z + 10z + 6, prepares the expression for combining.

Combining like terms involves adding or subtracting their coefficients. Since −z is equivalent to −1z, adding −1z + 10z yields 9z. Thus, the simplified expression becomes 9z + 6. This expression is now fully simplified, as it contains no parentheses and all like terms are combined.

Consider another example: 2a² − (6a² − b²) + 5b². The first step is to distribute the negative sign (which is equivalent to multiplying by −1) across the parentheses, changing the expression to 2a² − 6a² + b² + 5b². Identifying like terms, 2a² and −6a² are like terms, as are b² and 5b². Grouping them together, the expression becomes (2a² − 6a²) + (b² + 5b²). Combining coefficients, 2 − 6 = −4 and 1 + 5 = 6, resulting in the simplified expression −4a² + 6b².

By following these steps—distributing constants or variables, identifying and grouping like terms, and combining coefficients—you can simplify any algebraic expression. This process not only makes expressions easier to work with but also lays a strong foundation for solving more complex algebraic problems.

Problem

Simplify the following by combining like terms.
$\frac{1}{2} x + \frac{3}{4} x - \frac{1}{2} y + \frac{1}{2} y$

$\frac{5}{4} y$

$\frac{5}{4} x - y$

$\frac{5}{4} y - x$

$\frac{5}{4} x$

Problem

Simplify the following by combining like terms.
$2 m + 3 n - p + 4 m - 2 n + 5 p$

$6 m + n + 4 p$

$6 m + n - 4 p$

$4 m + n + 4 p$

$4 m - n - 4 p$

Problem

Simplify the expression.
$7 (x - 3) + 10$

$7 x minus 7$

$7 x plus 7$

$7 x minus 11$

$7 x plus 11$

Problem

Simplify the expression.
$6 (2 a - b) + 4 (3 a + 5 b)$

$24 a minus b$

$24 a plus b$

$24 a plus 14 b$

$24 a minus 14 b$

Problem

Simplify the expression.
$- 3 [2 x - (4 - x)]$

$- 9 x + 12$

$- 3 x + 12$

$9 x minus 12$

$3 x minus 12$

Problem

Simplify the expression.
$3 x^{2} + 5 x^{3} - 2 x + 4 x^{2} - x^{3} + 8 x + 10$

$x^{2} - 6 x^{3} + 9 x + 10$

$x^{2} + 6 x^{3} - 9 x + 10$

$7 x^{2} + 4 x^{3} + 6 x + 10$

$- 7 x^{2} + 4 x^{3} + 6 x + 10$

Problem

Simplify the expression.
$\frac{1}{2} x^{2} + \frac{3}{4} x y - \frac{1}{3} x^{2} + \frac{1}{4} x y$

$\frac{5}{6} x^{2} - x y$

$\frac{1}{6} x^{2} + x y$

$\frac{5}{6} x^{2} + x y$

$\frac{5}{6} x^{2} - \frac{1}{2} x y$

Here’s what students ask on this topic:

Like terms in algebraic expressions are terms that have the exact same variables raised to the same exponents. To identify like terms, you need to check that the variables match and that their exponents are identical. For example, $x^{2}$ and $x^{2}$ are like terms because both have the variable $x$ raised to the power of 2. However, $x^{2}$ and $x^{3}$ are not like terms since the exponents differ. Also, terms like $ab$ and $ba$ are like terms because multiplication order does not affect the variables involved. Identifying like terms is essential for combining them correctly during simplification.

To combine like terms, you add or subtract their coefficients while keeping the variable part unchanged. For example, if you have $4 x^{2} + 7 x^{2}$ , since both terms are like terms, you add the coefficients 4 and 7 to get $11 x^{2}$ . It is important to only combine terms that have the same variables with the same exponents. Unlike terms cannot be combined. This process simplifies expressions and makes them easier to work with in further algebraic operations.

Simplifying an algebraic expression involves several key steps: First, distribute any constants or variables across parentheses. For example, in $- z + 2 (3 + 5 z)$ , distribute the 2 to get $- z + 6 + 10 z$ . Second, identify like terms by checking variables and exponents. Third, group like terms together to prepare for combining. Finally, combine like terms by adding or subtracting their coefficients, resulting in a simplified expression with no parentheses and all like terms combined. This process makes expressions easier to understand and use in further calculations.

Unlike terms cannot be combined because they contain different variables or the same variables raised to different exponents. Combining terms requires that the variable parts match exactly, including their exponents. For example, $x^{2}$ and $x^{3}$ are unlike terms because the exponents differ, so their coefficients cannot be added or subtracted. Similarly, $a$ and $ab$ are unlike terms due to different variables. Attempting to combine unlike terms would violate algebraic rules and lead to incorrect results.

To simplify expressions with multiple variables and exponents, first distribute any constants or negatives across parentheses. Then, identify like terms by ensuring the variables and their exponents match exactly. For example, $5 x^{2 y z}$ and $- 10 x^{2 y z}$ are like terms because they have the same variables with the same exponents. Combine their coefficients to simplify. However, terms like $5 x^{2 y z}$ and $- 10 x^{y z^{2}}$ are not like terms due to different exponents on $y$ and $z$ . Simplifying requires careful attention to these details to combine terms correctly.

Your Beginning & Intermediate Algebra tutors

Simplifying Expressions: Videos & Practice Problems