Find the missing number so that the equation has no solutions.
3x–1+x=–
4x–19

To find the missing number so that the equation has no solutions, let's solve it step by step.

The given equation is:

\[
3x - 1 + x = -4x - 19
\]

### Step 1: Combine like terms on both sides.

On the left-hand side:

\[
3x + x - 1 = 4x - 1
\]

So the equation becomes:

\[
4x - 1 = -4x - 19
\]

### Step 2: Move all terms involving \( x \) to one side and constants to the other.

Add \( 4x \) to both sides:

\[
4x + 4x - 1 = -19
\]

This simplifies to:

\[
8x - 1 = -19
\]

### Step 3: Solve for \( x \).

Add 1 to both sides:

\[
8x = -18
\]

Now, divide by 8:

\[
x = -\frac{18}{8} = -\frac{9}{4}
\]

So the equation has a solution \( x = -\frac{9}{4} \).

### Step 4: Find the missing number for "no solutions".

In order for the equation to have no solutions, the variable terms must cancel out, and we are left with a false statement, like \( 5 = 3 \).

This means the coefficients of \( x \) on both sides must be the same, but the constants must differ.

To create a situation where the equation has no solutions, the missing number in the equation should make the constants unequal after cancelling out the \( x \)-terms. If we adjust the equation so that the missing number is **different from \(-1\)**, then the constants will not be equal.

Thus, the missing number could be **any value other than \(-1\)** to make the equation have no solutions.

---

Would you like a more detailed explanation of any step, or do you have any specific questions?

Here are 5 related questions you might find interesting:
1. How do you determine if an equation has infinitely many solutions?
2. What are the conditions for an equation to have no solutions?
3. What is the difference between equations with no solution and those with infinite solutions?
4. How do you solve an equation involving fractions?
5. Can an equation with variables in both denominators and numerators have no solution?

**Tip**: When simplifying equations, always combine like terms first to make the equation easier to work with!

Find the missing number so that the equation has no solutions. 3x - 1 + x = - 4x - 19

Solve the equation 3x - 1 + x = -4x - 19 step-by-step to find the missing number that results in no solution. Learn about algebraic concepts, including how coefficients and constants determine if an equation has no solutions. Suitable for Grades 8-10.

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