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Algebraic expressions

Algebra, grade 7 Lesson 11 of 22Learning

Algebraic expressions

An algebraic expression is made of numbers and variables joined by the operations of arithmetic, for example 3x+5\displaystyle 3x+5, 2(a−1)\displaystyle 2(a-1) or n4\displaystyle \frac{n}{4}. An algebraic expression is a short, exact way to write a “recipe” for a calculation.

From words to an expression
“5 more than” is addition: x+5\displaystyle x+5. “5 times” is multiplication: 5x\displaystyle 5x.

Call some number x\displaystyle x. Then the number 5 more than it is x+5\displaystyle x+5; the number 3 less than it is x−3\displaystyle x-3; the number 4 times as large is 4x\displaystyle 4x; half of it is x2\displaystyle \frac{x}{2}; its opposite is −x\displaystyle -x; and twice the number 1 more than it is 2(x+1)\displaystyle 2(x+1).

How algebra is written
Between two numbers the multiplication sign must be written: 3⋅5\displaystyle 3\cdot5, not 35.

Between a number and a variable, and between two variables, no multiplication sign is written: 3⋅x=3x\displaystyle {3\cdot x}={3x}, a⋅b=ab\displaystyle {a\cdot b}={ab}. The number is written before the variable: instead of x⋅3\displaystyle x\cdot3 we write 3x\displaystyle 3x. Instead of 1⋅x\displaystyle 1\cdot x we write x\displaystyle x, and instead of (−1)⋅x\displaystyle (-1)\cdot x we write −x\displaystyle -x.

From a story to an expression

A cinema ticket costs 35 ₪, and a bag of popcorn costs 20 ₪.
A family bought k\displaystyle k tickets and p\displaystyle p bags of popcorn.
How much did it pay?
35k\displaystyle 35k ₪ for the tickets and 20p\displaystyle 20p ₪ for the popcorn – 35k+20p\displaystyle 35k+20p ₪ in all.

Exercise 1

Write an algebraic expression: a. the number 7 more than y\displaystyle y; b. a third of y\displaystyle y; c. twice the sum of a\displaystyle a and b\displaystyle b; d. the number of days in w\displaystyle w weeks; e. Dana’s age in 4 years, if she is d\displaystyle d today.

Exercise 2

Noy has m\displaystyle m ₪.
She bought 3 notebooks, and each notebook costs c\displaystyle c ₪.
Write an expression for the money she has left.

The value of an expression
A negative number is substituted in brackets.

To find the value of an expression, substitute the numbers for the variables and calculate in the order of operations. The value of 2x−7\displaystyle 2x-7 for x=−3\displaystyle {x}={-3}: 2⋅(−3)−7=−6−7=−13\displaystyle {2\cdot(-3)-7}={-6-7}={-13}. And with two variables – the value of ab−a\displaystyle ab-a for a=4\displaystyle {a}={4} and b=−2\displaystyle {b}={-2}: 4⋅(−2)−4=−8−4=−12\displaystyle {4\cdot(-2)-4}={-8-4}={-12}.

Exercise 3

Find the value of each expression.
Start with the substitution.
a. 3x+4\displaystyle 3x+4 for x=−5\displaystyle {x}={-5}; b. 10−2y\displaystyle 10-2y for y=−4\displaystyle {y}={-4}; c. x2+1\displaystyle x^2+1 for x=−3\displaystyle {x}={-3}; d. 2a−3b\displaystyle 2a-3b for a=1\displaystyle {a}={1} and b=−2\displaystyle {b}={-2}.

a

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

    b

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

      / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

      c

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

        / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

        d

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

          / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

          Exercise 4

          Find the value of the expression 4−x\displaystyle 4-x for x\displaystyle x equal to −2\displaystyle -2, −1\displaystyle -1, 0\displaystyle 0, 1\displaystyle 1 and 2\displaystyle 2.
          What happens to the value of the expression as x\displaystyle x grows?

          x=−2\displaystyle {x}={-2}

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

            / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

            x=−1\displaystyle {x}={-1}

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

              / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

              x=0\displaystyle {x}={0}

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                x=1\displaystyle {x}={1}

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                  / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

                  x=2\displaystyle {x}={2}

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

                    / – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power