How many solutions does this equation have?

–

18

d

+

7

=

d

+

2

–

19

d

no solution

one solution

two solutions

infinitely many solutions

To determine how many solutions the equation has, let's simplify and solve it step by step.

The equation is:

\[
-18d + 7 = d + 2 - 19d
\]

### Step 1: Simplify both sides
First, combine like terms on the right side of the equation.

\[
-18d + 7 = (d - 19d) + 2
\]

\[
-18d + 7 = -18d + 2
\]

### Step 2: Move terms involving \( d \) to one side
Subtract \(-18d\) from both sides:

\[
7 = 2
\]

### Step 3: Analyze the result
The equation \( 7 = 2 \) is a contradiction, which is false. This means that there is no value of \( d \) that can satisfy the equation.

### Conclusion
The equation has **no solution**.

---

Would you like more details or have any other questions? Here are five related questions you might find interesting:

1. How do you determine if an equation has infinitely many solutions?
2. What does it mean for an equation to have no solution in terms of graphing?
3. How can you identify if an equation has one solution without solving it?
4. What types of equations always have two solutions?
5. What is the difference between a conditional equation and an identity?

**Tip:** Always simplify the equation as much as possible before solving it; this can make it easier to identify contradictions or special cases.

Explore how to determine the number of solutions for equations. Step-by-step solution provided for an example equation involving algebraic simplification and contradiction analysis. Suitable for Grades 7-9.

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