An algebraic expression combines numbers, variables, and operations, such as \(3n + 5\). A variable is a letter that stands for a number. To evaluate an expression, substitute a value for the variable and follow the order of operations. Equivalent expressions, like \(2(x+3)\) and \(2x+6\), have the same value for every \(x\).
Key facts:
- In \(7x + 2\), the terms are \(7x\) and \(2\), the coefficient is \(7\), and the constant is \(2\).
- Key words: sum means add, difference means subtract, product means multiply, quotient means divide.
- For a cube with edge \(s\), volume is \(V = s^3\) and surface area is \(A = 6s^2\).
- The distributive property says \(a(b + c) = ab + ac\), and like terms combine: \(5x + 2x = 7x\).
What is an algebraic expression?
An algebraic expression is a mathematical phrase made of numbers, variables, and operation symbols, such as \(3n + 5\). A variable is a letter that stands for a number that can change. Expressions have no equals sign, so they cannot be “solved,” only simplified or evaluated.
In \(3n + 5\), there are two terms: \(3n\) and \(5\). The number multiplying the variable, \(3\), is the coefficient. The term \(5\) has no variable and is a constant. In \(3n\), the numbers and letters multiplied together, \(3\) and \(n\), are factors.
How do you write an expression from words?
Match key words to operations: sum means add, difference means subtract, product means multiply, and quotient means divide.
| Words | Expression |
|---|---|
| 8 more than \(n\) | \(n + 8\) |
| 8 less than \(n\) | \(n – 8\) |
| the product of 4 and \(n\) | \(4n\) |
| \(n\) divided by 3 | \(\frac{n}{3}\) |
| twice the sum of \(n\) and 3 | \(2(n + 3)\) |
How do you evaluate an expression?
To evaluate, replace each variable with its given value and follow the order of operations. Example: evaluate \(3x^2 – 4\) when \(x = 5\). Substitute: \(3\times 5^2 – 4\). Exponent: \(3\times 25 – 4\). Multiply: \(75 – 4\). Subtract: \(71\).
Formulas are expressions too. A cube with edge length \(s\) has volume \(V = s^3\) and surface area \(A = 6s^2\). For \(s = 4\text{ in}\): \(V = 4^3 = 64\text{ in}^3\) and \(A = 6\times 16 = 96\text{ in}^2\).
What are equivalent expressions?
Equivalent expressions have the same value for every value of the variable. Two tools create them.
- Distributive property: \(a(b + c) = ab + ac\), so \(3(x + 4) = 3x + 12\).
- Combining like terms: terms with the same variable part can be added, so \(5x + 2x = 7x\).
Example: simplify \(4(2x + 3) + 5x – 6\).
- Distribute: \(8x + 12 + 5x – 6\).
- Group like terms: \((8x + 5x) + (12 – 6)\).
- Combine: \(13x + 6\).
Check by substituting \(x = 2\): the original gives \(4(7) + 10 – 6 = 32\), and \(13(2) + 6 = 32\). They match.
Writing and Evaluating Algebraic Expressions: frequently asked questions
What is the difference between an expression and an equation?
An equation has an equals sign and states that two quantities are equal, like \(x + 3 = 9\). An expression, like \(x + 3\), has no equals sign and is only a phrase that can be evaluated or simplified.
What is a coefficient?
A coefficient is the number multiplied by a variable. In \(8y + 5\), the coefficient of \(y\) is \(8\). If a variable has no number written in front, such as \(m\), its coefficient is \(1\).
How do you evaluate an expression?
Substitute the given number for each variable, then follow the order of operations. For \(2n + 7\) when \(n = 5\), compute \(2\times 5 + 7 = 17\).
What are like terms?
Like terms have the same variable part, so only their coefficients differ. For example, \(6a\) and \(2a\) are like terms and combine to \(8a\), but \(6a\) and \(2\) are not like terms.
How can you check that two expressions are equivalent?
Substitute the same value into both and compare. If \(3(x+4)\) and \(3x+12\) both equal \(21\) when \(x = 3\), that supports equivalence. One mismatch, however, proves they are not equivalent.
See every chapter on the Grade 6 math page.