Find the missing number so that the equation has **no** 

**solutions**.

2

(

5

x

–

20

)

+

x

+

8

=

–

4

(

x

–

11

)

Save answer




To solve this problem, we need to determine the conditions under which the equation has **no solutions**. Let's solve it step by step:

### Given equation:
\[
2(5x - 20) + x + 8 = -4(x - 11)
\]

### Step 1: Simplify both sides of the equation

**Left-hand side:**
\[
2(5x - 20) + x + 8 = 2 \cdot 5x - 2 \cdot 20 + x + 8 = 10x - 40 + x + 8 = 11x - 32
\]

**Right-hand side:**
\[
-4(x - 11) = -4 \cdot x + (-4) \cdot (-11) = -4x + 44
\]

### Step 2: Set the equation as:
\[
11x - 32 = -4x + 44
\]

### Step 3: Combine like terms

Move all the terms with \(x\) to one side and the constants to the other side:
\[
11x + 4x = 44 + 32
\]
\[
15x = 76
\]

### Step 4: Solve for \(x\)
\[
x = \frac{76}{15}
\]

This gives a solution, but for the equation to have **no solutions**, both sides of the equation must simplify to something contradictory (e.g., a false statement like \(5 = 3\)).

### Step 5: Analyze the condition for no solutions

For no solutions, the coefficients of \(x\) on both sides of the equation must cancel each other out, but the constants must not be equal. This means:
\[
\text{Coefficient of } x \text{ on the left-hand side} = \text{Coefficient of } x \text{ on the right-hand side}
\]

So, we want:
\[
11x = -4x
\]

If we move the \(x\)-terms together:
\[
11x + 4x = 0 \implies 15x = 0 \implies x = 0
\]

This tells us that for no solutions, there must be a contradiction. For that, let's adjust the constant terms accordingly.

**Would you like further details on how to finalize this contradiction?**

Here are 5 related questions to explore:
1. What does it mean for an equation to have infinite solutions instead of none?
2. How do you handle equations with variables on both sides?
3. What happens if both sides of an equation simplify to the same value?
4. Can a system of equations have no solutions? Why?
5. What is the difference between no solution and infinite solutions in an equation?

**Tip:** For an equation to have no solutions, aim for the variable terms to cancel out, leaving a false statement like \(5 = 3\).

Find the missing number so that the equation has no solutions. 2(5x - 20) + x + 8 = -4(x - 11)

This problem focuses on finding the missing number in a linear equation so that the equation has no solutions. The solution uses distribution, combining like terms, and the contradiction method. Suitable for Grades 8-10 students learning algebra.

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