Equations from real situations

Before you can solve an equation, you must first make one. The art of algebra begins with translating a situation , a balanced scale, a growing pattern of sticks, a price list , into an equality involving a letter-number.

Idea

An equation is two expressions joined by an equals sign: $\text{Left Hand Side (LHS)} = \text{Right Hand Side (RHS)}.$

Examples: $3x + 4 = 7$, $20 = y - 3$, $\dfrac{a}{3} = 50$, $2z + 4 = 5z - 14$.

To "solve" an equation means to find the value (or values) of the letter-number that make LHS equal to RHS.

From balance scales. A weighing scale is an equation in disguise. Suppose two sacks of unknown weight $s$ kg are balanced against one $10$ kg block and one $2$ kg block: $2s = 10 + 2 = 12.$ The fact that the scale is balanced gives us the equation. The letter $s$ stands for the unknown weight of one sack.

From matchstick patterns. Consider a row of triangles built from matchsticks:

• $1$ triangle uses $3$ sticks.
• $2$ triangles share a side and use $5$ sticks.
• $3$ triangles use $7$ sticks.

The pattern: the $n$-th arrangement uses $2n + 1$ sticks. If we have $99$ sticks and want to know how many triangles we can make: $2n + 1 = 99.$

From words. "I think of a number. Multiply it by $5$ and subtract $4$. The answer is $11$." Let the number be $x$: $5x - 4 = 11.$

Three rules of thumb when framing equations.

1. Name the unknown. Pick a letter , $x, y, n, e$ , for the thing you want to find.
2. Translate each English phrase into algebra. "twice", "double", "more than", "less than", "split equally" all have direct symbol equivalents.
3. Set the two sides equal. Use the fact (balance, total, equality) given by the problem.

Note. Different equations can describe the same situation. $2s + 2 = 12$ and $2s = 10$ are equally valid for the sack-and-block scale, depending on whether you keep all the weights on or remove the $2$ kg block.

Worked examples

Example 1. A balance scale has $3$ identical apples on one pan and a $150$ g weight balancing them. Frame an equation for the weight $a$ of one apple.

$3a = 150$.

Example 2. Two bread slices weigh $2$ g each. They balance two fried eggs of unknown equal weight $e$. Frame an equation.

LHS (bread side) $= 2 + 2 = 4$. RHS (eggs) $= e + e = 2e$. Equation: $2e = 4$.

Example 3. Squares are made with matchsticks: $1$ square uses $4$ sticks, two share a side and use $7$, three use $10$, … so the $n$-th uses $3n+1$ sticks. How many squares can be made with $46$ sticks?

Equation: $3n + 1 = 46$.

Example 4. "A number divided by $3$, then increased by $5$, gives $12$." Frame an equation.

Let the number be $x$. Equation: $\dfrac{x}{3} + 5 = 12$.

Try it yourself

1. Frame an equation: "Twice a number plus $7$ is $19$."
2. A balance shows $4$ bananas equal to $200$ g. Equation for weight of one banana?
3. A row of $n$ pentagons sharing sides uses $4n+1$ sticks. Frame an equation for $n$ if $33$ sticks are used.
4. "The sum of a number and $13$ is $25$." Equation?
5. "$3$ less than $5$ times a number equals $32$." Equation?
6. Two unknown weights $w$ on one pan are balanced by $3$ weights of $4$ kg each. Equation?
7. A taxi charges $\rupee 50$ as fixed fare plus $\rupee 12$ per km. Total fare is $\rupee 158$. Equation for distance $d$?
8. A girl is $x$ years old. Her father is $30$ years older. Their total age is $52$. Equation?

Activity

Make two stacks of identical school books on a small table. Add a known weight (water bottle, brick) to one stack so the two heights match. Write an equation that expresses the weight balance. Now solve it (intuition is fine!) and check.