Lesson 14: Evaluation and Solving Equations by Inspection

New Concept

Algebraic thinking begins by using letters, called variables, to represent unknown numbers. This lets us build expressions and equations to describe relationships and solve problems.

What’s next

Next, we'll put this idea into practice. You'll learn to evaluate expressions by substituting values and solve basic equations just by looking at them.

Property

To evaluate an expression, substitute the given numbers for the variables and calculate the result. This process turns a general algebraic formula into a specific, concrete answer.

Examples

• To evaluate $P = 2l + 2w$ when $l=10$ and $w=5$, you calculate $P = 2(10) + 2(5) = 20 + 10 = 30$.
• To find the value of the expression $3ab$ for $a = 2$ and $b = 5$, you calculate $3(2)(5)$, which equals 30.
• For $y = x - 7$, find the value of $y$ when $x = 20$. Solution: $y = 20 - 7 = 13$.

Explanation

It's like a video game where variables are empty item slots. When you're given numbers, you plug them into the slots and the game calculates your final score. Time to level up your math!

Property

An equation states that two quantities are equal. Solving by inspection means you mentally determine the variable's value that makes the equation true, without writing out formal steps.

Examples

• For the equation $w + 6 = 20$, you can inspect it and see that $w$ must be 14 because $14 + 6 = 20$.
• To solve $4x + 5 = 25$, you think: "Something plus 5 is 25, so $4x$ must be 20. That means $x$ must be 5."
• For $\frac{d}{3} = 9$, you ask yourself: "What number divided by 3 equals 9?" The answer is 27, so $d=27$.

Explanation

Be a math detective! Use your brain to look at the clues in the equation and deduce the mystery number that solves the puzzle. You're solving it just by looking at it—no calculator needed!

Property

A constant is a number whose value never changes, like 4 or -10. A variable is a letter, like x or s, that is used to represent a number whose value can change or is unknown.

Examples

• In the expression $7y + 2$, the numbers 7 and 2 are constants, while $y$ is the variable.
• In the taxi fare formula $3m + 2 = c$, the 3 (cost per mile) and 2 (flat fee) are constants. The letters $m$ (miles) and $c$ (cost) are variables.

Explanation

Imagine a constant is your birthday—the date is fixed! A variable is like the weather; it can be different every day. In algebra, numbers stand still, but letters represent values that can change.