Given the equation $$ 5 x+15=5 $$ select which of the following are equivalent. $$ 9 x+15=9 $$ $$ 10 x+30=10 $$ $$ x+15=1 $$ $$ 5 x+16=6 $$ $$ 5 x+2 \sqrt{5}=-5 $$

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UpStudy AI · Accepted Answer

1. Start with the given equation:
   $$
   5x + 15 = 5.
   $$
   
2. Subtract $$15$$ from both sides to isolate the term with $$x$$:
   $$
   5x = 5 - 15 = -10.
   $$
   
3. Divide both sides by $$5$$ to solve for $$x$$:
   $$
   x = \frac{-10}{5} = -2.
   $$

Now, an equation is equivalent to the given equation if it has the same solution $$x = -2$$.

---

**Checking each option:**

1. **Equation:** $$9x + 15 = 9$$

- Solve for $$x$$:
     $$
     9x = 9 - 15 = -6 \quad \Longrightarrow \quad x = -\frac{6}{9} = -\frac{2}{3}.
     $$
   - Since $$-\frac{2}{3} \neq -2$$, this equation is not equivalent.

2. **Equation:** $$10x + 30 = 10$$

- Solve for $$x$$:
     $$
     10x = 10 - 30 = -20 \quad \Longrightarrow \quad x = \frac{-20}{10} = -2.
     $$
   - Because $$x = -2$$ matches the solution of the given equation, this equation is equivalent.

3. **Equation:** $$x + 15 = 1$$

- Solve for $$x$$:
     $$
     x = 1 - 15 = -14.
     $$
   - Since $$-14 \neq -2$$, this equation is not equivalent.

4. **Equation:** $$5x + 16 = 6$$

- Solve for $$x$$:
     $$
     5x = 6 - 16 = -10 \quad \Longrightarrow \quad x = \frac{-10}{5} = -2.
     $$
   - Hence, this equation is equivalent.

5. **Equation:** $$5x + 2\sqrt{5} = -5$$

- Solve for $$x$$:
     $$
     5x = -5 - 2\sqrt{5} \quad \Longrightarrow \quad x = -\frac{5 + 2\sqrt{5}}{5}.
     $$
   - This $$x$$ value is not equal to $$-2$$ (since it would require $$2\sqrt{5} = 5$$, which is false). Therefore, this equation is not equivalent.

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**Conclusion:** The equations equivalent to $$5x + 15 = 5$$ are:
$$
10x + 30 = 10 \quad \text{and} \quad 5x + 16 = 6.
$$
The equivalent equations are $$10x + 30 = 10$$ and $$5x + 16 = 6$$.

Given the equation

select which of the following are equivalent.

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Given the equation select which of the following are equivalent.