Linear equations with a balance model

What is it about?

You can think of an equation such as $3x + 2 = x + 8$ as a balance in equilibrium: on the left there are three boxes, each weighing $x$ , and two unit weights of $1$ ; on the right one box and eight unit weights. Both sides weigh the same – that is what the equals sign means.

We are looking for the weight $x$ of one box. You change the balance so that it stays level until only one box is left on one side:

\begin{aligned} 3x + 2 &= x + 8 &&\mid -x\\ 2x + 2 &= 8 &&\mid -2\\ 2x &= 6 &&\mid :2\\ x &= 3 \end{aligned}

Transformations that do not change the solution set are called equivalent transformations:

add or subtract the same number or the same term on both sides,
multiply both sides by the same number or divide both sides by the same number (except $0$ ).

In the margin you note what you do to both sides, e.g. $\mid -x$ . The result is written as the solution set $\{3\}$ . You check it in the original equation: $3 \cdot 3 + 2 = 11$ and $3 + 8 = 11$ .

Special cases: If $x$ disappears completely, a statement without a variable remains. If it is false (e.g. $3 = 5$ ), the equation has no solution: $\{\,\}$ . If it is true (e.g. $4 = 4$ ), every number is a solution.

Limits of the model: Weights are always positive. Equations with a negative solution (e.g. $x + 9 = 3x + 13$ , so $x = -2$ ) or with minus signs in the terms cannot be shown with a balance. $2x = 3$ cannot be split further in the model either – by calculation, $x = 1.5$ .

Try it

Tap a box on the left only. What happens to the balance? How do you restore the balance?
Solve $3x + 2 = x + 8$ with the buttons Both sides − x, Both sides − 1 and Both sides : 2. Watch the chain of transformations on the right.
Open the boxes: how heavy is one? Does the check work?
Divide both sides of 4x + 2 = 2x + 6 by 2. Why does the balance stay level?
No solution? Remove all boxes on both sides. What is left?
Limit of the model: Why can the balance never be level here?
2x = 5: Why can’t the weights be split into two equal parts? Open the boxes: how heavy is one?
Press New equation for a random task.

Tasks

Task 1: Solve with the balance

Load task Solve $5x + 2 = 2x + 14$ . Write down the chain of transformations and check your answer.

Show solution

\begin{aligned} 5x + 2 &= 2x + 14 &&\mid -2x\\ 3x + 2 &= 14 &&\mid -2\\ 3x &= 12 &&\mid :3\\ x &= 4 \end{aligned}

Solution set $\{4\}$ . Check: $5 \cdot 4 + 2 = 22$ and $2 \cdot 4 + 14 = 22$ .

Task 2: The x is on the right

Load task Solve $x + 12 = 4x + 3$ .

Show solution

$x + 12 = 4x + 3 \mid -x$ gives $12 = 3x + 3 \mid -3$ , so $9 = 3x \mid :3$ and $3 = x$ . Solution set $\{3\}$ .

Task 3: From the picture to the equation

On the left there are 4 boxes and 1 weight, on the right 2 boxes and 9 weights; the balance is level. Set up the equation and find the weight of one box.

Show solution

$4x + 1 = 2x + 9 \mid -2x \Rightarrow 2x + 1 = 9 \mid -1 \Rightarrow 2x = 8 \mid :2 \Rightarrow x = 4$ . Show in the simulation

Task 4: Find the mistake

Max writes: $3x + 4 = x + 10 \mid -x$ gives $3x + 4 = 10$ . What did he do wrong? What does the balance show? Load the original equation

Show solution

Max removed an $x$ on the right only – the balance tips. Correct: $2x + 4 = 10 \mid -4 \Rightarrow 2x = 6 \mid :2 \Rightarrow x = 3$ .

Task 5: Limits of the model

Solve by calculation: (a) $2x + 7 = 4x + 3$ and (b) $x + 6 = 3x + 10$ . Which of the two equations can be solved with the balance?

Show solution

(a) $2x + 7 = 4x + 3 \mid -2x \Rightarrow 7 = 2x + 3 \mid -3 \Rightarrow 4 = 2x \mid :2 \Rightarrow x = 2$ . This also works on the balance.

(b) $x + 6 = 3x + 10 \mid -x \Rightarrow 6 = 2x + 10 \mid -10 \Rightarrow -4 = 2x \mid :2 \Rightarrow x = -2$ . A box with negative weight does not exist; even the step $\mid -10$ is impossible on the balance, because there are only 6 weights on the left. Show the balance

Task 6: Special cases

Find the solution set: (a) $3x + 5 = 3x + 2$ , (b) $2x + 2 = 2(x + 1)$ .

Show solution

(a) $\mid -3x$ gives $5 = 2$ – false, so the solution set is $\{\,\}$ . The balance tips to the left for every box weight.

(b) Expanding gives $2x + 2 = 2x + 2$ . $\mid -2x$ gives $2 = 2$ – always true, so every number is a solution. The balance is level for every box weight. The balance cannot show brackets, only the expanded form.

Ideas for the classroom

Use: About grade 7 (linear equations: setting up equations and solving them by equivalent transformations) – as a visual introduction before equations with negative numbers are solved purely by calculation.
How it works: A tap removes an object on one side only – the balance tips and the transformation is marked red. Only when the same is removed on the other side does a valid line appear (consecutive steps of the same kind are combined as in an exercise book, e.g. $\mid -2$ ). The buttons transform both sides; “Both sides : n” suggests a suitable divisor and visibly splits the pans into equal groups.
Shared links contain the original equation. “New equation” creates tasks with a natural-number solution from 1 to 6.
Limits of the model (discuss them deliberately): no negative numbers and no differences such as $3x - 2$ , positive solutions only, dividing only when it works out (for $2x = 5$ the simulation gives $x = 2.5$ as the calculated solution). Adding to and multiplying both sides are not offered here, because they are not needed to solve such equations. Mention them anyway: if you put the same on both pans, the balance also stays level.
Without mouse or finger: Single objects can only be removed by tapping or clicking. With the keyboard, the buttons for both sides are available.
Common stumbling blocks:
- Only one side is transformed (exactly what the tipping balance shows).
- When dividing, only part of a side is divided: $2x + 4 = 10 \mid :2$ does not give $x + 4 = 5$ but $x + 2 = 5$ .
- “Move it to the other side and change the sign” as a rule without understanding – the balance shows that in fact the same is subtracted on both sides.