Reasoning about Solving Equations (Part 2)

Student Summary

The balanced hanger diagram shows the amounts on the left equal the amounts on the right. The left side has 3 pieces that each have unknown weight xx and 3 pieces that each weigh 2 units. So, the left side shows 3 xx’s plus 6 units. The right side shows 18 units.  We could represent this diagram with an equation and solve the equation the same way we did before.

3x+6=183x=12x=4\begin{aligned} 3x+6&=18 \\ 3x&=12 \\ x&=4 \\ \end{aligned}

Balanced hanger. Left side, circle labeled x, square labeled 2, circle labeled x, square labeled 2, circle labeled x, square labeled 2. Right side, rectangle labeled 18.

Since there are 3 groups of x+2x+2 on the left, we could represent this hanger with a different equation: 3(x+2)=183(x+2)=18.

Balanced hanger, three groups are indicated, each group contains 1 circle labeled x and 1 square labeled 2. Right side, rectangle labeled 18.  To the side, an equation 3 ( x + 2 ) = 18.

The two sides of the hanger balance with these weights: 3 groups of x+2x+2 on one side, and 18, or 3 groups of 6, on the other side.

Balanced hanger. to the side, an equation.
Balanced hanger, left side, circle labeled x, square labeled 2, circle labeled x, square labeled 2, circle labeled x, square labeled 2, right side, rectangle not labeled.  A dotted line is drawn around three groups, each group contains one circle and one square from the left side and a third of the rectangle on the right side.  To the side, an equation says 1/3 * 3 ( x + 2 ) = 1/3 * 18.

The two sides of the hanger will balance with 13\frac13 of the weight on each side:

Balanced hanger, left side, 1 circle labeled x and 1 square labeled 2, right side, rectangle labeled 6.  To the side, an equation says x + 2 = 6.

We can remove 2 units of weight from each side, and the hanger will stay balanced. This is the same as subtracting 2 from each side of the equation.

Balanced hanger.
Balanced hanger. Left side, circle labeled x and square labeled 2, the square appears to be loose from the hanger. Right side, rectangle labeled 4 and square labeled 2, the square appears to be loose from the hanger.  To the side, an equation says x + 2 - 2= 6 - 2.

An equation for the new balanced hanger is x=4x=4. This gives the solution to the original equation.

Balanced hanger, left side, circle labeled x, right side, rectangle labeled 4. To the side, an equation x = 4.

Here is a concise way to write the steps above:

3(x+2)=18x+2=6after multiplying each side by 13x=4after subtracting 2 from each side\begin{aligned} 3(x+2) &= 18 \\ x + 2 &= 6 & \text{after multiplying each side by } \tfrac13 \\ x &= 4 & \text{after subtracting 2 from each side} \\ \end{aligned}

Visual / Anchor Chart