# Evaluating Algebraic Expressions Overview

### What is an algebraic expression?

An **algebraic expression** is a mathematical phrase that includes variables, constants, coefficients, and algebraic operations. For example, \(5x^2+6xy-c\) is an algebraic expression. Unlike algebraic equations, algebraic expressions do not have equal signs.

**Evaluating Algebraic Expressions Sample Questions**

There are several types of algebraic expressions. A **monomial** is an algebraic expression with only one term in which the exponents and variables are non-negative integers. A **binomial** is an algebraic expression composed of two monomials linked by an operation symbol. A **trinomial** consists of three monomials linked by operation symbols. A **polynomial** is an algebraic expression with any amount of terms greater than 1 but not an infinite amount.

### How do you evaluate algebraic expressions?

To evaluate algebraic expressions, substitute each variable’s value into the expression. When substituting a number in for a variable, it is always a good idea to put it in parentheses. This helps you correctly simplify exponents and multiply. Once you’ve done this, simplify using the **order of operations**.

For instance, to evaluate the expression \(4s^2-1\) when \(s=2\), substitute the number \(2\) into the expression for \(s\) and solve:

\(4(2)^2-1\) | First, substitute \(2\) into the expression for \(s\). |

\(4(4)-1\) | The expression involves multiplication, an exponent, and subtraction. According to the order of operations, simplify exponents before multiplying or subtracting. Since \(2\) raised to the second power equals \(4\), rewrite the expression using \(4\). |

\(16-1\) | The two operations remaining in the expression are multiplication and subtraction. According to the order of operations, multiply before subtracting. Since \(4\times4=16\), rewrite the expression using \(16\). |

\(15\) | Finally, subtract. Since \(16-1=15\), the expression \(4s^2-1\) is simplified to \(15\) when \(s=2\). |

### Examples:

- Evaluate \(xyz\) if \(x=3,y=2,\) and \(z=1\).
\((3)(2)(1)\) First, substitute \(3\) into the expression for \(x\), \(2\) for \(y\), and \(1\) for \(z\). When \(2\) or more variables are adjacent to each other with no operation sign, always multiply. In this case, multiply \(3\times2\times1\). \((6)(1)\) Multiply from left to right. Since \(3\times2=6\), rewrite the expression using \(6\). \(6\) Finally, multiply \((6)(1)\). Since \(6\times1=6\), the expression \(xyz\) is simplified to \(6\) when \(x=3,y=2,\) and \(z=1\). - Evaluate \(\frac{x+(3+y^3)}{10-z}\) if \(x=5,y=3,\) and \(z=3\).
\(\frac{(5)+(3+(3)^3)}{10-(3)}\) First, substitute \(5\) into the expression for \(x\), \(3\) for \(y\), and \(3\) for \(z\). \(\frac{(5)+(3+27)}{10 – (3)}\) This expression involves parentheses, exponents, division, addition, and subtraction. Start by addressing the portion of the expression in parentheses, \((3+(3)^3)\). According to the order of operations, simplify exponents first. Since \(3^3\) is equal to \(27\), rewrite the expression using \(27\). \(\frac{(5)+(30)}{10 – (3)}\) Next, continue to simplify the portion of the expression in parentheses by adding. Since \(3+27=30\), rewrite the expression using \(30\). \(\frac{35}{7}\) From here, simplify the numerator and denominator. Since \(5+30=35\), write \(35\) as the numerator. Since \(10-3=7\), write \(7\) as the denominator. \(5\) Finally, simplify the fraction by dividing the numerator by the denominator. Since \(35\div7=5\), the expression \(\frac{x+(3+y^3)}{10-z}\) is simplified to \(5\) when \(x=5,y=3,\) and \(z=3\). - Evaluate \(-3a-(2b+4)\) if \(a=-8\) and \(b=-7\).
\(-3(-8)-(2(-7)+4)\) First, substitute \(-8\) into the expression for \(a\) and \(-7\) for \(b\). \(-3(-8)-((-14)+4)\) This expression involves parentheses, multiplication, addition, and subtraction. Start by addressing the portion of the expression in parentheses, \((2(-7)+4)\). According to the order of operations, simplify \(2(-7)\) first. Since \(2-7=-14\), rewrite the expression using this product. \(-3(-8)-(-10)\) Next, continue to simplify the portion of the expression in parentheses by simplifying \(((-14)+4)\). Since \(-14+4=-10\), rewrite the expression using this sum. \(24-(-10)\) From here, multiply \(-3(-8)\). Since \(-3\times-8=24\), rewrite the expression using this product. \(34\) Finally, subtract \(24-(-10)\). Since \(24-(-10)=34\), the expression \(-3a-(2b+4)\) is simplified to \(34\) when \(a=-8\) and \(b=-7\).

**Evaluating Algebraic Expressions Sample Questions**

Here are a few sample questions going over evaluating algebraic expressions.

**Question #1:**

Evaluate \(\frac{ac}{b}\) if \(a=5,b=2,\) and \(c=10\).

**Answer:**

\(\frac{(5)(10)}{2}\) | First, substitute \(5\) into the expression for \(a,2\) for \(b\), and \(10\) for \(c\). |

\(\frac{50}{2}\) | From here, simplify the numerator. Since \(5\times10=50\), rewrite the expression with \(50\) as the numerator. |

\(25\) | Finally, simplify the fraction by dividing the numerator by the denominator. Since \(50\div2=25\), the expression can be simplified to \(25\). Therefore, the correct answer is C. |

**Question #2:**

Evaluate \(3xyz-8\) if \(x=-4,y=-3,\) and \(z=-6\).

**Answer:**

\(3(-4)(-3)(-6)-8\) | First, substitute \(-4\) into the expression for \(x\), \(-3\) for \(y\), and \(-6\) for \(z\). |

\(-12(-3)(-6)-8\) | From here, simplify the expression using the order of operations. Start by multiplying from left to right, beginning with \(3(-4)\). Since \(3\times-4=-12\), rewrite the expression using \(-12\). |

\(36(-6)-8\) | Next, multiply \(-12\) by \(-3\). Since \(-12\times-3=36\), rewrite the expression using \(36\). |

\(-216-8\) | Continue simplifying by multiplying \(36\) by \(-6\). Since \(36\times-6=-216\), rewrite the expression using \(-216\). |

\(-224\) | Finally, subtract. Since \(-216-8=-224\), the expression can be simplified to \(-224\). Therefore, the correct answer is C. |

**Question #3:**

Evaluate \(\frac{3(a + c^2)}{12}\) if \(a=-9\) and \(c=5\).

**Answer:**

\(\frac{3((-9)+(5)^2)}{12}\) | First, substitute \(-9\) into the expression for \(a\) and \(5\) for \(c\). |

\(\frac{3((-9) + 25)}{12}\) | From here, simplify the expression using the order of operations. Start with the portion in parentheses, which is \(((-9)+(5)^2)\). Simplify exponents first. Since \(5^2=25\), rewrite the expression using \(25\). |

\(\frac{3(16)}{12}\) | Next, continue simplifying the portion of the expression in parentheses by adding \(-9\) and \(25\). Since \(-9+25=16\), rewrite the expression using \(16\). |

\(\frac{48}{12}\) | From here, simplify the numerator. Since \(3\times16=48\), rewrite the expression using \(48\). |

\(4\) | Finally, simplify the fraction by dividing the numerator by the denominator. Since \(48\div12=4\), the expression can be simplified to \(4\). Therefore, the correct answer is D. |

**Question #4:**

Jack and Diane own a company that sells scooters. The amount they pay the employees on their sales team (in dollars) is \(15x+25y\). In this expression, \(x\) represents the number of hours worked and \(y\) represents the number of sales made. How much would an employee make if they work for \(5\) hours and sell \(4\) scooters?

**Answer:**

\(15(5)+25(4)\) | First, substitute \(5\) into the expression for \(x\) and \(4\) for \(y\). |

\(75+100\) | From here, simplify the expression using the order of operations. Since we must multiply before adding, simplify each multiplication portion of the expression. \(15\times5=75\), and \(25\times4=100\). Rewrite the expression using these products. |

\(175\) | Finally, add \(75\) and \(100\). Since \(75+100=175\), the expression can be simplified to \(175\). At the scooter company, an employee would make \($175\) if they worked for \(5\) hours and sold \(4\) scooters. Therefore, the correct answer is B. |

**Question #5:**

Claire’s bakery sells croissants, muffins, and donuts. The expression \(3.5c+4m+2.5d\) gives the cost (in dollars) of \(c\) croissants, \(m\) muffins, and \(d\) donuts. Will is having a party and needs to order \(10\) croissants, \(6\) muffins, and \(12\) donuts. What is the total cost of Will’s order?

**Answer:**

\(3.5(10)+4(6)+2.5(12)\) | First, substitute \(10\) into the expression for \(c\), \(6\) for \(m\), and \(12\) for \(d\). |

\(35+24+30\) | From here, simplify the expression using the order of operations. Since we must multiply before adding, simplify each multiplication portion of the expression. \(3.5\times10=35\), \(4\times6=24\), and \(2.5\times12=30\). Rewrite the expression using these products. |

\(89\) | Finally, add \(35,24,\) and \(30\). Since \(35+24+30=89\), The expression can be simplified to \(89\). Will’s bakery order will cost \(89\) dollars if he buys \(10\) croissants, \(6\) muffins, and \(12\) donuts. Therefore, the correct answer is D. |