Evaluating Expressions: Videos & Practice Problems

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.

Evaluating Expressions

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, coefficients are fixed numbers that multiply the variables, and constants are numbers that stand alone without any attached variables. 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 their values, coefficients and constants remain fixed. This distinction helps in evaluating algebraic expressions, which means substituting the variables with specific numerical values and then performing the arithmetic operations.

To evaluate an algebraic expression, replace each variable with the given value and simplify using the order of operations. For instance, evaluating \)2x + 5\( when \)x = 4\( involves substituting 4 for \)x\(, resulting in \)2 \times 4 + 5\(. Calculating this gives \)8 + 5 = 13\(.

When expressions contain multiple variables, such as \)\frac{1}{2}a + 4b\(, you substitute each variable with its corresponding value. For example, if \)a = 10\( and \)b = -6\(, the expression becomes \)\frac{1}{2} \times 10 + 4 \times (-6)\(, which simplifies to \)5 - 24 = -19\(.

Exponents in algebraic expressions indicate repeated multiplication of a base number. For example, evaluating \)-8y^3\( when \)y = 2\( requires calculating \)2^3 = 2 \times 2 \times 2 = 8\(, then multiplying by \)-8\( to get \)-8 \times 8 = -64$.

Mastering algebraic expressions involves recognizing variables, coefficients, and constants, and confidently evaluating expressions by substituting values and applying arithmetic operations correctly. This foundational skill is essential for progressing in algebra and solving more complex mathematical problems.

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

• A. $6$
• B. $2$
• C. $-2$
• D. $12$

Answer: (B)

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

• A. $9$
• B. $-2$
• C. $0$
• D. $-9$

Answer: (D)

Evaluating Expressions Example 1

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

First, calculate the square of -3, which is 9, since squaring a negative number yields a positive result. Then multiply by 2:

\)2 \times 9 = 18\(

Next, multiply 5 by 4:

\)5 \times 4 = 20\(

Finally, add these two results together to find the value of the expression:

\)18 + 20 = 38$

This process demonstrates how to evaluate algebraic expressions by substituting variables with given values and performing arithmetic operations step-by-step. Understanding this method is essential for solving a wide range of algebra problems efficiently and accurately.

Evaluating Expressions Example 2

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

\(-5 - 3 \times 2^2\)

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

\$2^2 = 4\(

Then multiply this result 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.

Evaluating Expressions Example 3

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 -3/2 and y equals 6, consider the expression 4x + 2/3. Substituting x with -3/2 transforms the expression into \$4 \times \left(-\frac{3}{2}\right) + \frac{2}{3}\(. Multiplying 4 (which can be written as \)\frac{4}{1}\() by \)-\frac{3}{2}\( involves multiplying numerators and denominators: \)4 \times (-3) = -12\( and \)1 \times 2 = 2\(, resulting in \)-\frac{12}{2}\(. Simplifying this fraction yields \)-6\(, so the expression becomes \)-6 + \frac{2}{3}\(. Combining these terms requires a common denominator, so rewrite \)-6\( as \)-\frac{18}{3}\(, leading to \)-\frac{18}{3} + \frac{2}{3} = -\frac{16}{3}\(. This fraction cannot be simplified further, so the final value is \)-\frac{16}{3}\(.

For a more complex expression such as \)2x - 0.5y + 20\(, substitute x with \)-\frac{3}{2}\( and y with 6, resulting in \)2 \times \left(-\frac{3}{2}\right) - 0.5 \times 6 + 20\(. Multiplying \)2\( (or \)\frac{2}{1}\() by \)-\frac{3}{2}\( gives \)-\frac{6}{2}\(, which simplifies to \)-3\(. The term \)0.5 \times 6\( equals 3, so the expression becomes \)-3 - 3 + 20\(. Combining these values, \)-3 - 3\( equals \)-6\(, and adding 20 results in 14. Therefore, the evaluated expression equals 14.

These examples highlight the importance of understanding fraction multiplication, simplification, and combining like terms when evaluating algebraic expressions. Recognizing equivalent fractions and decimals, such as 0.5 being \)\frac{1}{2}$, can simplify calculations. Mastery of these skills enables accurate and efficient evaluation of expressions for given variable values.

Evaluating Expressions Example 4

To evaluate the algebraic expression 2x + 3y - z² when given specific values for the variables, substitute each variable with its corresponding value. 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\(

Since squaring a negative number results in a positive value, calculate \)(-2)^2\( as:

\)(-2) \times (-2) = 4\(

Substitute this back into the expression:

\)2 + 12 - 4\(

Adding and subtracting in order, first add 2 and 12:

\)14 - 4\(

Then subtract 4 from 14 to get the final result:

\)10$

This process demonstrates how to evaluate expressions with multiple variables by systematically substituting values and applying arithmetic operations, reinforcing the foundational skills of algebraic manipulation and order of operations.

Here’s what students ask on this topic:

What does it mean to evaluate an algebraic expression?

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 $\mathrm{2x}+5$ and you are asked to evaluate it for $x=4$, you replace $x$ with 4, resulting in $2\times 4+5$. Then, you 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.

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 operations. For example, consider the expression $\frac{1}{2}a+4b$ with $a=10$ and $b=-6$. Replace $a$ and $b$ to get $\frac{1}{2}\times 10+4\times -6$. 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.

What is the role of coefficients and constants in algebraic expressions?

In algebraic expressions, coefficients are the numerical factors that multiply the variables, and they remain constant. For example, in $2x$, 2 is the coefficient. Constants are numbers without variables and also remain unchanged, such as the 5 in $2x+5$. While the variable's value can change, coefficients and constants do not. Understanding these parts helps in evaluating expressions correctly by identifying which parts to substitute and which to keep fixed.

How do you evaluate expressions with exponents?

When evaluating expressions with exponents, first substitute the variable with the given value, then apply the exponent before performing other operations. For example, consider $-8y{y}^3$ with $y=2$. Replace $y$ to get $-8\times 2^3$. Calculate the exponent first: $2^3=8$. Then multiply: $-8\times 8=-64$. This order ensures accurate evaluation of expressions involving powers.

What is the difference between variables, coefficients, and constants?

Variables are letters representing values that can change, such as $x$, $y$, or $z$. Coefficients are fixed numbers multiplying the variables, like the 2 in $2x$. Constants are fixed numbers without variables, such as 5 in $2x+5$. While variables vary, coefficients and constants remain the same. Recognizing these parts helps in correctly evaluating and simplifying algebraic expressions.