Answer :
We start with the equation
[tex]$$
-15x + 7 \cdot 15 - 15 = 3^7 \cdot 9^{-3} \cdot x.
$$[/tex]
Step 1. Simplify the left-hand side
Calculate the constant part:
1. Multiply [tex]$7$[/tex] by [tex]$15$[/tex]:
[tex]$$
7 \cdot 15 = 105.
$$[/tex]
2. Subtract [tex]$15$[/tex] from [tex]$105$[/tex]:
[tex]$$
105 - 15 = 90.
$$[/tex]
Thus, the left-hand side becomes:
[tex]$$
-15x + 90.
$$[/tex]
Step 2. Simplify the right-hand side
1. Calculate [tex]$3^7$[/tex]:
[tex]$$
3^7 = 2187.
$$[/tex]
2. Calculate [tex]$9^{-3}$[/tex]. Recall that a negative exponent represents the reciprocal:
[tex]$$
9^{-3} = \frac{1}{9^3} = \frac{1}{729}.
$$[/tex]
3. Multiply these two results:
[tex]$$
2187 \cdot \frac{1}{729} = 3.
$$[/tex]
Thus, the right-hand side becomes:
[tex]$$
3 \cdot x \quad \text{or} \quad 3x.
$$[/tex]
Step 3. Set up the simplified equation
Now the equation is:
[tex]$$
-15x + 90 = 3x.
$$[/tex]
Step 4. Solve for [tex]$x$[/tex]
1. Add [tex]$15x$[/tex] to both sides to collect like terms:
[tex]$$
-15x + 90 + 15x = 3x + 15x \quad \Longrightarrow \quad 90 = 18x.
$$[/tex]
2. Divide both sides by [tex]$18$[/tex]:
[tex]$$
x = \frac{90}{18} = 5.
$$[/tex]
Final Answer:
The solution to the equation is
[tex]$$
x = 5.
$$[/tex]
[tex]$$
-15x + 7 \cdot 15 - 15 = 3^7 \cdot 9^{-3} \cdot x.
$$[/tex]
Step 1. Simplify the left-hand side
Calculate the constant part:
1. Multiply [tex]$7$[/tex] by [tex]$15$[/tex]:
[tex]$$
7 \cdot 15 = 105.
$$[/tex]
2. Subtract [tex]$15$[/tex] from [tex]$105$[/tex]:
[tex]$$
105 - 15 = 90.
$$[/tex]
Thus, the left-hand side becomes:
[tex]$$
-15x + 90.
$$[/tex]
Step 2. Simplify the right-hand side
1. Calculate [tex]$3^7$[/tex]:
[tex]$$
3^7 = 2187.
$$[/tex]
2. Calculate [tex]$9^{-3}$[/tex]. Recall that a negative exponent represents the reciprocal:
[tex]$$
9^{-3} = \frac{1}{9^3} = \frac{1}{729}.
$$[/tex]
3. Multiply these two results:
[tex]$$
2187 \cdot \frac{1}{729} = 3.
$$[/tex]
Thus, the right-hand side becomes:
[tex]$$
3 \cdot x \quad \text{or} \quad 3x.
$$[/tex]
Step 3. Set up the simplified equation
Now the equation is:
[tex]$$
-15x + 90 = 3x.
$$[/tex]
Step 4. Solve for [tex]$x$[/tex]
1. Add [tex]$15x$[/tex] to both sides to collect like terms:
[tex]$$
-15x + 90 + 15x = 3x + 15x \quad \Longrightarrow \quad 90 = 18x.
$$[/tex]
2. Divide both sides by [tex]$18$[/tex]:
[tex]$$
x = \frac{90}{18} = 5.
$$[/tex]
Final Answer:
The solution to the equation is
[tex]$$
x = 5.
$$[/tex]
