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. For example, combining $4x^2 + 7x^2$ results in $11 x^2$ . Simplification enhances understanding of polynomials, monomials, and multivariable polynomials by focusing on coefficients, exponents, and the structure of terms, essential for mastering algebraic manipulation and polynomial degree concepts.

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Like Terms

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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 all variables and their exponents match exactly.

It is important to note that coefficients—the numerical factors in terms—do not affect whether terms are like terms. Terms are like terms solely based on their variables and exponents.

On the other hand, 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). Terms like 2a and 2ba are not like terms because the variables differ. Another example is 5x²yz and -10xy²z, which are unlike terms because 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 because the coefficients 4 and 7 are added. Similarly, 2ab + 8ba can be combined as 10ab after recognizing the terms as like terms and rearranging variables if needed.

It is crucial to remember that unlike terms cannot be combined by adding or subtracting their coefficients. For example, 5x²yz - 10xy²z cannot be simplified further because the terms are not like terms due to differing exponents.

Understanding how to identify terms, recognize like terms, and combine them correctly is essential for simplifying algebraic expressions and solving equations efficiently.

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Like Terms Example 1

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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. Like terms can be combined by adding or subtracting their coefficients while keeping the variable and its exponent unchanged.

For example, in the expression 6a² + 8a², both terms contain the variable a raised to the power of 2. Since the variables and exponents match, these are like terms. Adding their coefficients, 6 and 8, results in 14a².

In contrast, consider the expression 7x² + 8x + 4. Here, the first term has x squared, while the second term has x to the first power (implied exponent 1). Because the exponents differ, these are not like terms and cannot be combined. The constant term 4 stands alone and also cannot be combined with the others.

Another example is -5y⁵ + 3y⁵. Both terms have the variable y raised to the fifth power, making them like terms. Combining their coefficients, -5 and 3, gives:

\[-5y^5 + 3y^5 = (-5 + 3) y^5 = -2y^5\]

Understanding how to identify and combine like terms is fundamental in simplifying algebraic expressions, which is a key skill in algebra and higher-level mathematics.

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Like Terms Example 2

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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 like terms, add their coefficients: 3, -9, and 1 (since p is the same as 1p). Calculating this gives:

\[3 - 9 + 1 = -5\]

So, the combined term is -5p.

Next, consider the constants in the expression, which are 4 and -2. Constants are like terms because they have no variable (or equivalently, the variable is raised to the zero power). Adding these constants yields:

\[4 - 2 = 2\]

Therefore, after combining all like terms, the simplified expression is:

\[-5p + 2\]

This process of combining like terms helps to simplify expressions and makes them easier to work with in further algebraic operations.

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Like Terms Example 3

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Like Terms Example 3 Video Summary

When simplifying algebraic expressions, combining like terms is essential to ensure each variable appears only once. Consider the expression 6z - 4 + 11z + 9. To combine like terms, first identify terms with the same variable and exponent. Here, 6z and 11z both contain the variable z raised to the first power (implied exponent of 1). These are like terms, so their coefficients can be added: \$6 + 11 = 17$. This results in $17z\(.

Next, focus on the constant terms, which are numbers without variables. The constants in this expression are \)-4$ and $9$. Since they are like terms (both constants), they can be combined by addition: $-4 + 9 = 5\(.

Putting it all together, the simplified expression is \)17z + 5$, where the variable z appears only once, and the constants are combined into a single term. This process of combining like terms is fundamental in algebra for simplifying expressions and solving equations efficiently.

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.
$0.5 z + .25 z - 1.35 + 1.55$

$0.75 z minus 0.2$

$0.75 z plus 0.2$

$0.95 z$

$negative 2.15 z$

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Simplify Expressions

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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 eliminates parentheses and prepares the expression for combining like terms.

Next, identify like terms, which are terms containing 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 considered separately since they have no variable component.

After identifying like terms, group them together by writing them adjacent to each other. This organization simplifies the next step, which is combining like terms by adding or subtracting their coefficients. For instance, \)-z + 10z$ can be combined by recognizing the coefficient of $-z$ as $-1$, so $-1 + 10 = 9$, resulting in $9z$. The constant 6 remains unchanged, giving the fully simplified expression $9z + 6\(.

Consider another example: \)2a^2 - (6a^2 - b^2) + 5b^2$. Here, the negative sign before the parentheses acts as $-1$ and must be distributed inside, changing the expression to $2a^2 - 6a^2 + b^2 + 5b^2$. Identifying like terms, $2a^2$ and $-6a^2$ are like terms with variable $a$ squared, while $b^2$ and $5b^2$ are like terms with variable $b$ squared. Combining these, $2a^2 - 6a^2$ equals $-4a^2$, and $b^2 + 5b^2$ equals $6b^2$. The simplified expression is $-4a^2 + 6b^2$.

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

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

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

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

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

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Simplify Expressions Example 4

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Simplify Expressions Example 4 Video Summary

To simplify an algebraic expression by expanding and combining like terms, the first step involves distributing any constants or variables across the terms inside parentheses. For example, when distributing 3x across the terms 2x² and -5xy, multiply the coefficients and then apply the laws of exponents to the variables. Multiplying 3x by 2x² results in 6x³ because 3 times 2 equals 6, and x times x² equals x³. Similarly, distributing 3x to -5xy gives -15x²y, since 3 times -5 is -15, and x times x is x², with y remaining as is.

Next, distribute -4y across x² and -2xy. Multiplying -4y by x² yields -4x²y. When multiplying -4y by -2xy, the product is 8xy² because the negatives cancel out, and y times y becomes y². It is important to maintain variables in alphabetical order for clarity, so terms are written as x²y rather than yx².

After distributing, identify like terms by comparing variables and their exponents. Terms such as -15x²y and -4x²y are like terms because they contain the same variables raised to the same powers. However, terms like 6x³ and 8xy² do not have like terms to combine with, as their variable exponents differ.

Combining like terms involves adding or subtracting their coefficients while keeping the variable part unchanged. For instance, -15x²y and -4x²y combine to form -19x²y. The final simplified expression after distribution and combination is:

\[6x^{3} - 19x^{2}y + 8xy^{2}\]

This process highlights the importance of careful distribution, attention to signs, and proper identification of like terms to simplify algebraic expressions effectively.

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 simplifying expressions by combining them through addition or subtraction of their coefficients.

To combine like terms, first identify terms with the same variables and exponents. Then, add or subtract their coefficients while keeping the variable part unchanged. For example, combining $4 x^{2} + 7 x^{2}$ results in $11 x^{2}$ because you add the coefficients 4 and 7. Remember, coefficients are the numerical parts of terms, and they can be positive or negative. You cannot combine unlike terms, such as $x^{2}$ and $x^{3}$ , because their exponents differ. Combining like terms simplifies expressions and makes them easier to work with in further algebraic operations.

Simplifying algebraic expressions involves several key steps. First, distribute any constants or variables outside parentheses to each term inside the parentheses. For example, in $- z + 2 (3 + 5 z)$ , distribute 2 to get $- z + 6 + 10 z$ . Second, identify like terms—terms with the same variables and exponents. Third, group these like terms together by writing them next to each other. Finally, combine the like terms by adding or subtracting their coefficients. The result is a fully simplified expression with no parentheses and all like terms combined, making it easier to understand and use in further calculations.

Unlike terms cannot be combined because they have different variables or the same variables raised to different exponents. Combining terms requires that the variable parts match exactly, including their exponents, to ensure the operation is mathematically valid. For example, $x^{2}$ and $x^{3}$ cannot be combined because the powers of $x$ differ. Similarly, $a$ and $ab$ are unlike terms due to the extra variable $b$ in the second term. Attempting to combine unlike terms would violate algebraic rules and lead to incorrect results. Therefore, only like terms can be added or subtracted by combining their coefficients.

Coefficients are the numerical factors in terms of algebraic expressions. When simplifying expressions, coefficients are added or subtracted when combining like terms. For example, in the terms $4 x^{2}$ and $7 x^{2}$ , the coefficients 4 and 7 are combined to get 11, resulting in $11 x^{2}$ . Coefficients do not affect whether terms are like or unlike; only the variables and their exponents determine that. However, coefficients determine the magnitude and sign of the combined term after simplification, playing a crucial role in the final simplified expression.

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Simplifying Expressions: Videos & Practice Problems