Which value or expression can you write in the blank so the equation has no solution? 0 13 $$ 6 y+13=6 y $$

Question

UpStudy AI · Accepted Answer

Let's consider the equation in the form

$$
a y + 13 = 6y.
$$

We want to choose the value for $$ a $$ (the blank) so that the equation has no solution.

### Step 1. Isolate the terms containing $$ y $$

Subtract $$ 6y $$ from both sides:

$$
a y - 6y + 13 = 0.
$$

### Step 2. Factor the $$ y $$

$$
(a - 6)y + 13 = 0.
$$

### Step 3. Determine condition for no solution

A linear equation of the form

$$
by + c = 0
$$

has no solution if the coefficient of $$ y $$ is zero (i.e., $$ b = 0 $$) but the constant $$ c $$ is nonzero, since then the equation becomes

$$
0 \cdot y + c = 0 \quad \Longrightarrow \quad c = 0,
$$

which is false if $$ c \neq 0 $$.

In our equation, we have:

$$
b = a - 6 \quad \text{and} \quad c = 13.
$$

For the equation to have no solution, we need

$$
a - 6 = 0 \quad \text{and} \quad 13 \neq 0.
$$

Since $$ 13 \neq 0 $$ is already true, we set

$$
a - 6 = 0.
$$

### Step 4. Solve for $$ a $$

$$
a = 6.
$$

### Conclusion

By writing $$ 6 $$ in the blank (i.e., taking $$ a=6 $$), the equation becomes

$$
6y+13=6y,
$$

which simplifies as follows:

$$
6y + 13 - 6y = 13 = 0.
$$

This is a contradiction because $$ 13 \neq 0 $$. Therefore, the equation has no solution.

**Answer:** $$6$$
Write $$6$$ in the blank so the equation has no solution.

Which value or expression can you write in the blank so the equation has no solution? 0 13 \( 6 y+13=6 y \)