2. Expressions & 3D Geometry

Lesson

In algebra we are very often required to substitute specific values into variables and constants.

The great philosopher and mathematician Rene Descartes (1596-1650) decided to use letters at the front of the alphabet for constants and letters at the back of the alphabet for variables. Generally speaking, this convention is still in use today.

In algebra, these variables to represent unknown values. For example if we have the equation $x+2=5$`x`+2=5, then we can work out that $x=3$`x`=3 since we also know that $3+2=5$3+2=5.

Sometimes we want to do this process in reverse, however, and we substitute numbers into equations in place of variables to determine a final value. We can substitute any numbers including whole numbers, decimals, and fractions.

When we substitute a number into an expression and find its value we call this evaluating the expression for the given value.

If $x=3$`x`=3, evaluate $6x-4$6`x`−4.

**Think: **This means that everywhere the letter $x$`x` has been written, we will replace it with the number $3$3.

**Do:**

$6x-4$6x−4 |
$=$= | $6\times3-4$6×3−4 | substituting $3$3 for $x$x |

$=$= | $18-4$18−4 | evaluating the multiplication | |

$=$= | $14$14 | evaluating the subtraction |

The same process applies even if there is more than one unknown value.

If $x=6$`x`=6 and $y=0.5$`y`=0.5, evaluate $6x-2y-12$6`x`−2`y`−12.

**Think:** Just like before, we will replace the letter $x$`x` with the number $6$6, and the letter $y$`y` with the number $0.5$0.5. We also need to keep the order of operations in mind when we do these kinds of calculations!

**Do: **

$6x-2y-12$6x−2y−12 |
$=$= | $6\times6-2\times0.5-12$6×6−2×0.5−12 | replacing $x$x with $6$6, and $y$y with $0.5$0.5 |

$=$= | $36-1-12$36−1−12 | evaluating multiplication | |

$=$= | $23$23 | evaluating the subtraction |

Now let's try some practice questions.

Evaluate $8x+4$8`x`+4 when $x=2$`x`=2.

Evaluate $6x+4y+6$6`x`+4`y`+6 when $x=3$`x`=3 and $y=5$`y`=5.

Evaluate $\frac{4a\times5}{9b}$4`a`×59`b` when $a=7$`a`=7 and $b=21$`b`=21.

Find the exact value in simplest form.

Write, read, and evaluate expressions in which letters stand for numbers.

Evaluate expressions at specific values for their variables. Include expressions that arise from formulas in real-world problems. Perform arithmetic operations, including those involving whole-number exponents, in the conventional order when there are no parentheses to specify a particular order (order of operations).