Question

1. match each set of equations with a possible step that turns the first equation into the second equation.

equations \t\t\t\tpossible steps
a \t$9x + 3 = 3x + 18$
\t$3x + 1 = x + 6$ \t\tmultiply each side by $-3$.
b \t$-3(5x - 7) = -21$
\t\t$5x - 7 = 7$ \t\tdivide each side by $-3$.
c \t$-\frac{4}{3}x = 5$
\t\t$4x = -15$ \t\tdivide each side by $3$.
d \t$7x + 5 = 3x - 6$
\t\t$4x + 5 = -6$ \t\tsubtract $3$ from each side.
e \t$10 - 5x = 8 + 4x$
\t\t$7 - 5x = 5 + 4x$ \t\tsubtract $3x$ from each side.

1. in this balanced hanger diagram, the weight of the triangle is $x$ and the weight of the circle is $y$. which equation using $x$ and $y$ represents the hanger diagram?

a. $3y + 2x = y + 3x$
b. $3x + 2y = x + 3y$
c. $2x + 3y = 3x + y$
d. $5xy = 4xy$

1. jada and tyler were each trying to solve $7x - 4 = 5x - 5$.

describe the first step they each made to the equation.
a \tthe result of jada’s first step was $7x = 5x - 1$.

Explanation:

Problem 1

Step1: Match set a to step

Take $9x + 3 = 3x + 18$, divide each term by 3:
$\frac{9x+3}{3} = \frac{3x+18}{3}$ simplifies to $3x + 1 = x + 6$.

Step2: Match set b to step

Take $-3(5x - 7) = -21$, divide each side by $-3$:
$\frac{-3(5x-7)}{-3} = \frac{-21}{-3}$ simplifies to $5x - 7 = 7$.

Step3: Match set c to step

Take $-\frac{4}{3}x = 5$, multiply each side by $-3$:
$-3 \times (-\frac{4}{3}x) = 5 \times (-3)$ simplifies to $4x = -15$.

Step4: Match set d to step

Take $7x + 5 = 3x - 6$, subtract $3x$ from each side:
$7x + 5 - 3x = 3x - 6 - 3x$ simplifies to $4x + 5 = -6$.

Step5: Match set e to step

Take $10 - 5x = 8 + 4x$, subtract 3 from each side:
$10 - 5x - 3 = 8 + 4x - 3$ simplifies to $7 - 5x = 5 + 4x$.

Step1: Count weights per side

Left side: 3 triangles ($x$) + 2 circles ($y$) = $3x + 2y$
Right side: 1 triangle ($x$) + 3 circles ($y$) = $x + 3y$

Step2: Set sides equal (balanced hanger)

Balanced hanger means left = right: $3x + 2y = x + 3y$

Step1: Analyze Jada's first step

Original equation: $7x - 4 = 5x - 5$
Result: $7x = 5x - 1$
To get from original to result, add 4 to both sides:
$7x - 4 + 4 = 5x - 5 + 4$ simplifies to $7x = 5x - 1$.

Answer:

a. Divide each side by 3.
b. Divide each side by -3.
c. Multiply each side by -3.
d. Subtract 3x from each side.
e. Subtract 3 from each side.

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Problem 2