Question

1
drag each tile to the correct location on the table. each tile can be used more than once, but not all tiles will be used.
choose the justification for each step in the solution to the given equation.
addition property of equality subtraction property of equality simplification
division property of equality multiplication property of equality
step justification
\\(\frac{17}{3} - \frac{3}{4}x = \frac{1}{2}x + 5\\) given
\\(\frac{17}{3} - \frac{3}{4}x - \frac{17}{3} = \frac{1}{2}x + 5 - \frac{17}{3}\\)
\\(-\frac{3}{4}x = \frac{1}{2}x - \frac{2}{3}\\)
\\(-\frac{3}{4}x - \frac{1}{2}x = \frac{1}{2}x - \frac{2}{3} - \frac{1}{2}x\\)
\\(-\frac{5}{4}x = -\frac{2}{3}\\)
\\(-\frac{5}{4}x \cdot -\frac{4}{5} = -\frac{2}{3} \cdot -\frac{4}{5}\\)
\\(x = \frac{8}{15}\\)
reset next

Explanation:

To solve this, we analyze each step and its corresponding justification:

Step 1: $\boldsymbol{\frac{17}{3} - \frac{3}{4}x - \frac{17}{3} = \frac{1}{2}x + 5 - \frac{17}{3}}$

We subtract $\frac{17}{3}$ from both sides. This matches the subtraction property of equality (subtracting the same value from both sides preserves equality).

Step 2: $\boldsymbol{-\frac{3}{4}x = \frac{1}{2}x - \frac{2}{3}}$

Simplify both sides by combining like terms ($\frac{17}{3} - \frac{17}{3} = 0$ and $5 - \frac{17}{3} = \frac{15}{3} - \frac{17}{3} = -\frac{2}{3}$). This is simplification.

Step 3: $\boldsymbol{-\frac{3}{4}x - \frac{1}{2}x = \frac{1}{2}x - \frac{2}{3} - \frac{1}{2}x}$

We subtract $\frac{1}{2}x$ from both sides. This matches the subtraction property of equality (subtracting the same term from both sides preserves equality).

Step 4: $\boldsymbol{-\frac{5}{4}x = -\frac{2}{3}}$

Simplify both sides by combining like terms ($-\frac{3}{4}x - \frac{1}{2}x = -\frac{3}{4}x - \frac{2}{4}x = -\frac{5}{4}x$ and $\frac{1}{2}x - \frac{1}{2}x = 0$). This is simplification.

Step 5: $\boldsymbol{-\frac{5}{4}x \cdot -\frac{4}{5} = -\frac{2}{3} \cdot -\frac{4}{5}}$

We multiply both sides by $-\frac{4}{5}$ (the reciprocal of $-\frac{5}{4}$). This matches the multiplication property of equality (multiplying both sides by the same non - zero value preserves equality).

Step 6: $\boldsymbol{x = \frac{8}{15}}$

Simplify both sides by multiplying the fractions ($-\frac{5}{4} \cdot -\frac{4}{5} = 1$ and $-\frac{2}{3} \cdot -\frac{4}{5} = \frac{8}{15}$). This is simplification.

Final Table of Justifications:

Step	Justification
$-\frac{3}{4}x = \frac{1}{2}x - \frac{2}{3}$	simplification
$-\frac{3}{4}x - \frac{1}{2}x = \frac{1}{2}x - \frac{2}{3} - \frac{1}{2}x$	subtraction property of equality
$-\frac{5}{4}x = -\frac{2}{3}$	simplification
$-\frac{5}{4}x \cdot -\frac{4}{5} = -\frac{2}{3} \cdot -\frac{4}{5}$	multiplication property of equality
$x = \frac{8}{15}$	simplification

Answer:

To solve this, we analyze each step and its corresponding justification:

Step 1: $\boldsymbol{\frac{17}{3} - \frac{3}{4}x - \frac{17}{3} = \frac{1}{2}x + 5 - \frac{17}{3}}$

We subtract $\frac{17}{3}$ from both sides. This matches the subtraction property of equality (subtracting the same value from both sides preserves equality).

Step 2: $\boldsymbol{-\frac{3}{4}x = \frac{1}{2}x - \frac{2}{3}}$

Simplify both sides by combining like terms ($\frac{17}{3} - \frac{17}{3} = 0$ and $5 - \frac{17}{3} = \frac{15}{3} - \frac{17}{3} = -\frac{2}{3}$). This is simplification.

Step 3: $\boldsymbol{-\frac{3}{4}x - \frac{1}{2}x = \frac{1}{2}x - \frac{2}{3} - \frac{1}{2}x}$

We subtract $\frac{1}{2}x$ from both sides. This matches the subtraction property of equality (subtracting the same term from both sides preserves equality).

Step 4: $\boldsymbol{-\frac{5}{4}x = -\frac{2}{3}}$

Simplify both sides by combining like terms ($-\frac{3}{4}x - \frac{1}{2}x = -\frac{3}{4}x - \frac{2}{4}x = -\frac{5}{4}x$ and $\frac{1}{2}x - \frac{1}{2}x = 0$). This is simplification.

Step 5: $\boldsymbol{-\frac{5}{4}x \cdot -\frac{4}{5} = -\frac{2}{3} \cdot -\frac{4}{5}}$

We multiply both sides by $-\frac{4}{5}$ (the reciprocal of $-\frac{5}{4}$). This matches the multiplication property of equality (multiplying both sides by the same non - zero value preserves equality).

Step 6: $\boldsymbol{x = \frac{8}{15}}$

Simplify both sides by multiplying the fractions ($-\frac{5}{4} \cdot -\frac{4}{5} = 1$ and $-\frac{2}{3} \cdot -\frac{4}{5} = \frac{8}{15}$). This is simplification.

Final Table of Justifications:

Step	Justification
$-\frac{3}{4}x = \frac{1}{2}x - \frac{2}{3}$	simplification
$-\frac{3}{4}x - \frac{1}{2}x = \frac{1}{2}x - \frac{2}{3} - \frac{1}{2}x$	subtraction property of equality
$-\frac{5}{4}x = -\frac{2}{3}$	simplification
$-\frac{5}{4}x \cdot -\frac{4}{5} = -\frac{2}{3} \cdot -\frac{4}{5}$	multiplication property of equality
$x = \frac{8}{15}$	simplification