Question

Arrange the tiles on both boards to model the equation:

2x + 4 = 3x + (-1). Then complete the sentences:

To model 2x + 4, you need:
a) Positive x-tiles to represent 2x,
b) Positive unit tiles to represent 4.

To model 3x + (-1), you'll need:
a) 3 x-tiles to represent 3x,
b) A unit tile to represent -1.

Board sum: 0

Answer 1

The equation
`\( x = x \)` is true, so the board sum is 0, and the equation is balanced.

To model the algebraic equation
`\( 2x + 4 = 3x - 1 \)` using tiles on both sides of an equation, we're essentially creating a balance between the two sides. Each tile represents a term in the equation.

Let's break it down:

For the left side of the equation,
`\( 2x + 4 \)`:

- We use 2 positive
`\( x \)`-tiles to represent
`\( 2x \)`.

- We use 4 positive unit tiles to represent
`\( +4 \)`.

For the right side of the equation,
`\( 3x - 1 \)`:

- We use 3 positive
`\( x \)-`tiles to represent
`\( 3x \)`.

- We use 1 negative unit tile to represent
`\( -1 \)`.

Now, we want to balance the equation by arranging the tiles on both boards. We should have an equal number of
`\( x \)`-tiles and unit tiles on both sides to make the board sum equal to 0, which means the equation is balanced.

Here are the steps to balance the equation using tiles:

1. We start by placing the tiles as per the initial representation of the equation:

- Left board: 2 positive
`\( x \)`-tiles, 4 positive unit tiles.

- Right board: 3 positive
`\( x \)`-tiles, 1 negative unit tile.

2. To balance the boards, we can remove a positive \( x \)-tile from each side. This is allowed because we are keeping the equation balanced by doing the same operation on both sides.

- Left board now has: 1 positive
`\( x \)`-tile, 4 positive unit tiles.

- Right board now has: 2 positive
`\( x \)`-tiles, 1 negative unit tile.

3. Next, we can also remove 4 positive unit tiles from the left side and 4 negative unit tiles from the right side to balance the units. Since we only have 1 negative unit tile on the right side, we can't remove 4, but we can add 3 more negative unit tiles to the right side to then remove 4 from each side.

4. After adding 3 more negative unit tiles to the right side (to make it possible to remove 4), we can remove 4 unit tiles from each side:

- Left board now has: 1 positive
`\( x \)`-tile.

- Right board now has: 2 positive
`\( x \)`-tiles, now 4 negative unit tiles (1 original and 3 added).

5. Finally, we remove 4 unit tiles from each side:

- Left board has: 1 positive
`\( x \)`-tile.

- Right board has: 2 positive
`\( x \)`-tiles, 4 negative unit tiles removed leaves us with 0 unit tiles.

6. The board is now unbalanced because the left board has 1 \( x \)-tile and the right board has 2
`\( x \)`-tiles. To balance it, we can add a negative \( x \)-tile to the left board and remove a positive
`\( x \)`-tile from the right board. This is like subtracting
`\( x \)` from both sides of the equation.

After these steps, we end up with the following configuration:

- Left board: 1 positive
`\( x \)`-tile added and 1 negative
`\( x \)`-tile subtracted.

- Right board: 2 positive \( x \)-tiles subtracted.

Now, we can write the complete sentences:

To model
`\( 2x + 4 \)`, you need:

a) 2 positive
`\( x \)`-tiles to represent
`\( 2x \)`,

b) 4 positive unit tiles to represent
`\( +4 \)`.

To model
`\( 3x - 1 \)`, you'll need:

a) 3 positive
`\( x \)`-tiles to represent
`\( 3x \)`,

b) 1 negative unit tile to represent
`\( -1 \)`.

After simplifying, the tiles on each board would represent:

- Left board:
`\( x \)` (since we subtracted an
`\( x \)`-tile from both sides).

- Right board:
`\( x \)` (since we also subtracted an
`\( x \)`-tile from the right side).

The equation
`\( x = x \)` is true, so the board sum is 0, and the equation is balanced.