How do you evaluate an expression with multiple variables?

To evaluate an expression with multiple variables, you substitute each variable with its given value and then perform the arithmetic operations. For example, consider the expression 1 2 a + 4 b with a = 10 and b = - 6 . Replace a and b with their values: 1 2 × 10 + 4 × - 6 . Calculate each part: 1 2 × 10 = 5 and 4 × - 6 = - 24 . Then add: 5 + - 24 = - 19 . This method works regardless of how many variables are present.

1. Review of Real Numbers

Evaluating Expressions

1. Review of Real Numbers

Evaluating Expressions: Videos & Practice Problems

Video Lessons

Topic summary

Algebraic expressions combine numbers, variables, and operations, where variables like $x$ , $y$ , or $z$ represent changing values. Coefficients are constant numerical factors multiplying variables, while constants are fixed numbers without variables. Evaluating expressions involves substituting variables with given values and applying operations, including exponents, as in $y^{3}$ . Understanding terms, coefficients, and exponents is essential for working with polynomials, monomials, and multivariable polynomials, laying the foundation for mastering concepts like degree, standard form, and scientific notation.

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Evaluating Expressions

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Evaluating Expressions Video Summary

Algebraic expressions are mathematical phrases that combine numbers and variables using operations such as addition, subtraction, multiplication, and division. A variable is a letter that represents an unknown or changeable value, commonly denoted by letters like x, y, or z. Unlike variables, a coefficient is a fixed number multiplying the variable, and a constant is a standalone number without any variable attached. For example, in the expression \$2x + 5$, the number 2 is the coefficient of the variable $x\(, and 5 is the constant.

Understanding the difference between variables, coefficients, and constants is crucial because while variables can change values, coefficients and constants remain fixed. This distinction helps when working with algebraic expressions, especially when evaluating them for specific values of the variables.

To evaluate an algebraic expression, substitute the given values for the variables and then perform the arithmetic operations following the order of operations (PEMDAS: Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). For instance, evaluating the expression \)2x + 5$ for $x = 4$ involves replacing $x$ with 4, resulting in $2 \times 4 + 5$. Calculating this gives $8 + 5 = 13\(.

When an expression contains multiple variables, such as \)\frac{1}{2}a + 4b$, you substitute each variable with its given value. For example, if $a = 10$ and $b = -6$, the expression becomes $\frac{1}{2} \times 10 + 4 \times (-6)$. Simplifying, $\frac{1}{2} \times 10 = 5$ and $4 \times (-6) = -24$, so the expression evaluates to $5 - 24 = -19\(.

Exponents are also important in algebraic expressions. For example, to evaluate \)-8y^3$ when $y = 2$, first calculate the exponent: $2^3 = 2 \times 2 \times 2 = 8$. Then multiply by the coefficient: $-8 \times 8 = -64$. Thus, the expression evaluates to $-64$.

Mastering algebraic expressions involves recognizing variables, coefficients, and constants, and confidently substituting values to evaluate expressions. This foundational skill is essential for solving equations and understanding more complex algebraic concepts.

Problem

Evaluate the following algebraic expression for the given value of $x$ .
$x^{2} + 4 x + 6; x = - 2$

$6$

$2$

$negative 2$

$12$

Problem

Evaluate the following algebraic expression for the given value of $x$ .
$7 x^{3} + 2 x - 9; x = 0$

$9$

$negative 2$

$0$

$negative 9$

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Evaluating Expressions Example 1

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Evaluating Expressions Example 1 Video Summary

To evaluate the algebraic expression 2a² + 5b given specific values for a and b, substitute the provided numbers directly into the expression. For example, if a = -3 and b = 4, replace every instance of a with -3 and every instance of b with 4. This results in the expression:

\$2(-3)^2 + 5(4)\(

Next, calculate the powers and products. Squaring -3 gives 9, so:

\)2 \times 9 + 5 \times 4 = 18 + 20\(

Adding these values yields:

\)18 + 20 = 38$

Thus, the value of the expression 2a² + 5b when a = -3 and b = 4 is 38. This process demonstrates how to evaluate algebraic expressions by substituting variables with given values and performing arithmetic operations step-by-step.

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Evaluating Expressions Example 2

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Evaluating Expressions Example 2 Video Summary

To evaluate the expression a − 3b² for given values of a and b, substitute the values directly into the expression. For example, if a = −5 and b = 2, replace a with −5 and b with 2, resulting in:

$-5 - 3 \times 2^2$

Next, calculate the exponent first, as per the order of operations. Squaring 2 gives:

\$2^2 = 4\(

Then multiply by 3:

\)3 \times 4 = 12\(

Now, subtract this product from −5:

\)-5 - 12 = -17$

Thus, the value of the expression a − 3b² when a = −5 and b = 2 is −17. This process highlights the importance of correctly applying substitution and following the order of operations, especially handling exponents before multiplication and subtraction.

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Evaluating Expressions Example 3

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Evaluating Expressions Example 3 Video Summary

When evaluating algebraic expressions with given values for variables, it is essential to substitute the values accurately and simplify step-by-step. For example, if x equals negative three halves and y equals six, consider the expression 4x + 2/3. Substituting x = -\frac{3}{2} transforms the expression into 4 \times -\frac{3}{2} + \frac{2}{3}. Multiplying fractions involves multiplying numerators and denominators directly, so 4 \times -\frac{3}{2} becomes \frac{4}{1} \times \frac{-3}{2} = \frac{-12}{2}, which simplifies to -6. Adding the remaining fraction yields -6 + \frac{2}{3}. Combining these terms requires expressing the integer as a fraction with a common denominator: -6 = \frac{-18}{3}, so the sum is \frac{-18}{3} + \frac{2}{3} = \frac{-16}{3}. This fraction cannot be simplified further, so the final value of the expression is -\frac{16}{3}.

For a more complex expression such as 2x - 0.5y + 20, substituting x = -\frac{3}{2} and y = 6 gives 2 \times -\frac{3}{2} - 0.5 \times 6 + 20. Multiplying 2 \times -\frac{3}{2} simplifies directly to -3 because the twos cancel out. The term 0.5 \times 6 equals 3, so the expression becomes -3 - 3 + 20. Combining these values, -3 - 3 = -6, and adding 20 results in 14. Therefore, the evaluated expression equals 14.

Understanding how to substitute values and simplify expressions involving fractions and decimals is crucial in algebra. Multiplying fractions requires multiplying numerators and denominators separately, while simplifying fractions involves dividing numerator and denominator by their greatest common divisor. Additionally, converting decimals to fractions or recognizing decimal equivalents (such as 0.5 being one-half) can streamline calculations. These skills enable accurate evaluation of expressions and reinforce foundational algebraic concepts.

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Evaluating Expressions Example 4

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Evaluating Expressions Example 4 Video Summary

To evaluate the algebraic expression 2x + 3y - z² when the variables are given specific values, substitute each variable with its corresponding number. For example, if x = 1, y = 4, and z = -2, replace each variable in the expression accordingly. This results in:

\$2 \times 1 + 3 \times 4 - (-2)^2\(

Next, perform the multiplication operations:

\)2 + 12 - (-2)^2\(

Recall that squaring a negative number involves multiplying the number by itself, which yields a positive result. Therefore, \)(-2)^2 = (-2) \times (-2) = 4\(. Incorporating this, the expression becomes:

\)2 + 12 - 4\(

Adding and subtracting in sequence, first add 2 and 12 to get 14, then subtract 4:

\)14 - 4 = 10$

Thus, the value of the expression 2x + 3y - z² when x = 1, y = 4, and z = -2 is 10. This process demonstrates how to handle algebraic expressions with multiple variables by substituting values and carefully applying arithmetic operations, including the correct evaluation of exponents involving negative numbers.

Here’s what students ask on this topic:

Evaluating an algebraic expression means substituting the variables in the expression with given numerical values and then performing the arithmetic operations to find the expression's value. For example, if you have the expression $2x + 5$ and you are asked to evaluate it for $x = 4$ , you replace $x$ with 4, resulting in $2 \times 4 + 5$ . Then, perform the multiplication and addition to get $8 + 5 = 13$ . This process helps you find the numerical value of the expression for specific variable values.

In algebraic expressions, coefficients and constants serve different roles. A coefficient is a fixed number that multiplies a variable. For example, in $2 x$ , 2 is the coefficient of the variable $x$ . Coefficients remain constant and do not change when evaluating the expression. On the other hand, a constant is a number without any variable attached, such as the 5 in $2x + 5$ . Constants also remain unchanged regardless of the variable's value. Understanding these terms is essential because when evaluating expressions, you substitute values only for variables, while coefficients and constants stay the same.

To evaluate an expression with multiple variables, you substitute each variable with its given value and then perform the arithmetic operations. For example, consider the expression $\frac{1}{2} a + 4 b$ with $a = 10$ and $b = - 6$ . Replace $a$ and $b$ with their values: $\frac{1}{2} \times 10 + 4 \times - 6$ . Calculate each part: $\frac{1}{2} \times 10 = 5$ and $4 \times - 6 = - 24$ . Then add: $5 + - 24 = - 19$ . This method works regardless of how many variables are present.

Exponents indicate repeated multiplication of a base number or variable. When evaluating expressions with exponents, you first calculate the power before performing other operations. For example, in the expression $- 8 y y^{3}$ evaluated at $y = 2$ , you calculate $2^{3}$ first, which is $2 \times 2 \times 2 = 8$ . Then multiply by the coefficient: $- 8 \times 8 = - 64$ . Understanding the order of operations, especially handling exponents first, is crucial for accurate evaluation.

Variables in algebraic expressions represent unknown or changing values. They are typically denoted by letters such as $x$ , $y$ , or $z$ . The value of a variable can vary, which means the expression can represent many different values depending on what the variable is assigned. For example, in the expression $2 x + 5$ , $x$ is the variable. When you evaluate the expression, you substitute $x$ with a specific number to find the expression's value. Variables allow algebraic expressions to model real-world situations where quantities change.