Question
How many solutions does this equation have? \[ -12+15 b=-3(-5 b+4) \]
Ask by Bond Christensen. in Ecuador
Jan 14,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
There are infinitely many solutions; any real number \( b \) satisfies the equation.
Solution
Let's analyze the given equation:
\[
-12 + 15b = -3(-5b + 4)
\]
**Step 1: Simplify the Right Side**
First, distribute the \(-3\) on the right side:
\[
-3(-5b + 4) = 15b - 12
\]
**Step 2: Rewrite the Equation**
Now, the equation becomes:
\[
-12 + 15b = 15b - 12
\]
**Step 3: Compare Both Sides**
Subtract \(15b\) from both sides:
\[
-12 = -12
\]
This simplifies to a true statement that holds for all real numbers \(b\).
**Conclusion:**
Since the equation simplifies to a true statement irrespective of the value of \(b\), there are **infinitely many solutions**. In other words, **any real number** \(b\) satisfies the equation.
**Answer:** There are infinitely many real solutions; every real number b satisfies the equation.
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Simplify this solution Extra Insights
To find the number of solutions for the equation \( -12 + 15b = -3(-5b + 4) \), we first simplify both sides. Starting with the right side: \[ -3(-5b + 4) = 15b - 12 \] Now our equation becomes: \[ -12 + 15b = 15b - 12 \] Next, we subtract \( 15b \) from both sides: \[ -12 = -12 \] This equation is always true, which means the original equation has infinitely many solutions. The variable \( b \) can take any value, resulting in a slew of possible solutions. It's like having an entire line of solutions rather than just isolated points! Thus, the equation simplifies to a true statement, indicating that it's valid for any \( b \).
