Question

drag justifications into the table to explain each step in solving the equation \\(\frac{3}{4}(x - 3) = \frac{5}{8}x + \frac{5}{4}x\\).

step	justification
\\(\frac{3}{4}x - \frac{9}{4} = \frac{5}{8}x + \frac{5}{4}x\\)
\\(6x - 18 = 5x + 10x\\)
\\(6x - 18 = 15x\\)
\\(-18 = 9x\\)
\\(-2 = x\\)

options for justification: combine like terms, distributive property, addition property of equality, multiplication property of equality, symmetric property, substitution property

Explanation:

Step1: Identify the first step's justification

The first step after the given equation is $\frac{3}{4}x - \frac{9}{4} = \frac{5}{8}x + \frac{5}{4}x$. This is obtained by applying the distributive property to $\frac{3}{4}(x - 3)$, which distributes $\frac{3}{4}$ to both $x$ and $-3$. So the justification here is "distributive property".

Step2: Identify the second step's justification

The next step is $6x - 18 = 5x + 10x$. To get from $\frac{3}{4}x - \frac{9}{4} = \frac{5}{8}x + \frac{5}{4}x$ to this, we multiply each term by 8 (the least common denominator of 4 and 8) to eliminate the fractions. This is the multiplication property of equality, which states that if we multiply both sides of an equation by the same non - zero number, the equation remains true. So the justification here is "multiplication property of equality".

Step3: Identify the third step's justification

The step $6x - 18 = 15x$ is obtained by combining the like terms $5x$ and $10x$ on the right - hand side. So the justification here is "combine like terms".

Step4: Identify the fourth step's justification

The step $- 18=9x$ is obtained by subtracting $6x$ from both sides of the equation $6x - 18 = 15x$. This is an application of the addition property of equality (subtraction is a form of addition of the opposite). So the justification here is "addition property of equality".

Step5: Identify the fifth step's justification

The step $- 2=x$ is obtained by dividing both sides of the equation $-18 = 9x$ by 9. This is the multiplication property of equality (division is a form of multiplication by the reciprocal). So the justification here is "multiplication property of equality" (or we can also think of it as dividing both sides by 9, which is allowed by the multiplication property of equality). And the last step, $-2 = x$ can also be written as $x=-2$ by the symmetric property, but in the step where we go from $-18 = 9x$ to $x=-2$, the main operation is division (multiplication by $\frac{1}{9}$), so the justification is "multiplication property of equality" (or division property, which is a subset of multiplication property).

Answer:

Step	Justification
$\frac{3}{4}x-\frac{9}{4}=\frac{5}{8}x+\frac{5}{4}x$	distributive property
$6x - 18=5x + 10x$	multiplication property of equality
$6x - 18 = 15x$	combine like terms
$-18=9x$	addition property of equality
$-2=x$	multiplication property of equality