Question

1. in this hanger, the weight of the triangle is x and the weight of the square is y.

a. write an equation using x and y to represent the hanger.
b. if x is 6, what is y?

1. match each set of equations with the move that turned the first equation into the second.
2. 6x + 9 = 4x - 3

2x + 9 = - 3
a. multiply both sides by \\( \frac { - 1 } { 4 } \\)

1. - 4(5x - 7) = - 18

5x - 7 = 4.5
b. multiply both sides by - 4

1. 8 - 10x = 7 + 5x

4 - 10x = 3 + 5x
c. multiply both sides by \\( \frac { 1 } { 4 } \\)

1. \\( \frac { - 5 x } { 4 } = 4 \\)

5x = - 16
d. add - 4x to both sides

1. 12x + 4 = 20x + 24

3x + 1 = 5x + 6
e. add - 4 to both sides

Explanation:

1.

a.

Since the hanger is balanced, the total weight on the left - hand side is equal to the total weight on the right - hand side.
The left - hand side has a weight of \(x + 3y\) (one triangle of weight \(x\) and three squares of weight \(y\) each).
The right - hand side has a weight of \(3x + y\) (three triangles of weight \(x\) each and one square of weight \(y\)).
So the equation is \(x + 3y=3x + y\).

b.

Step1: Substitute \(x = 6\) into the equation

Substitute \(x = 6\) into the equation \(x + 3y=3x + y\).
We get \(6+3y=3\times6 + y\).

Step2: Simplify the right - hand side

Simplify \(3\times6 + y\) to \(18 + y\). So the equation is \(6+3y=18 + y\).

Step3: Subtract \(y\) from both sides

Subtract \(y\) from both sides: \(6+3y - y=18 + y - y\).
This gives \(6 + 2y=18\).

Step4: Subtract 6 from both sides

Subtract 6 from both sides: \(6+2y-6 = 18-6\).
We have \(2y=12\).

Step5: Divide both sides by 2

Divide both sides by 2: \(y=\frac{12}{2}\).

2.

1.

Starting with \(6x + 9=4x-3\), if we add \(-4x\) to both sides:
\((6x + 9)+(-4x)=(4x - 3)+(-4x)\).
\(6x-4x + 9=4x-4x-3\), which simplifies to \(2x + 9=-3\).
So \(1\) matches with \(d\).

2.

Starting with \(-4(5x - 7)=-18\), if we multiply both sides by \(\frac{-1}{4}\) (or divide both sides by \(-4\)):
\(\frac{-4(5x - 7)}{-4}=\frac{-18}{-4}\).
\(5x-7 = 4.5\).
So \(2\) matches with \(a\).

3.

Answer:

1.
a. \(x + 3y=3x + y\)
b. \(y = 6\)
2.

1. \(d\)
2. \(a\)
3. \(e\)
4. \(b\)
5. \(c\)