Sure, let's solve each of the equations step by step.
### 1. [tex]\(3.12 \times a + 5 = 3 \times a - 1\)[/tex]
To isolate [tex]\(a\)[/tex]:
[tex]\[3.12a + 5 = 3a - 1\][/tex]
First, we move all terms involving [tex]\(a\)[/tex] to one side and constants to the other side:
[tex]\[3.12a - 3a = -1 - 5\][/tex]
[tex]\[0.12a = -6\][/tex]
Next, solve for [tex]\(a\)[/tex] by dividing both sides by 0.12:
[tex]\[a = \frac{-6}{0.12} = -50\][/tex]
So, the solution for [tex]\(a\)[/tex] is:
[tex]\[a = -50\][/tex]
### 2. [tex]\(3.27 \times b + 8 = 5 \times b + 12\)[/tex]
To isolate [tex]\(b\)[/tex]:
[tex]\[3.27b + 8 = 5b + 12\][/tex]
First, move all terms involving [tex]\(b\)[/tex] to one side and constants to the other side:
[tex]\[3.27b - 5b = 12 - 8\][/tex]
[tex]\[-1.73b = 4\][/tex]
Next, solve for [tex]\(b\)[/tex] by dividing both sides by -1.73:
[tex]\[b = \frac{4}{-1.73} \approx -2.31213872832370\][/tex]
So, the solution for [tex]\(b\)[/tex] is approximately:
[tex]\[b \approx -2.312\][/tex]
### 3. [tex]\(3 \times c^2 - 6 \times c = 0\)[/tex]
To solve for [tex]\(c\)[/tex]:
[tex]\[3c^2 - 6c = 0\][/tex]
Factorize the equation:
[tex]\[3c(c - 2) = 0\][/tex]
Set each factor to zero and solve:
[tex]\[3c = 0 \quad \text{or} \quad c - 2 = 0\][/tex]
[tex]\[c = 0 \quad \text{or} \quad c = 2\][/tex]
So, the solutions for [tex]\(c\)[/tex] are:
[tex]\[c = 0 \quad \text{or} \quad c = 2\][/tex]
### 4. [tex]\(3.43 \times d^2 = 6 \times d\)[/tex]
To isolate [tex]\(d\)[/tex]:
[tex]\[3.43d^2 = 6d\][/tex]
First, move all terms to one side of the equation:
[tex]\[3.43d^2 - 6d = 0\][/tex]
Factorize the equation:
[tex]\[d(3.43d - 6) = 0\][/tex]
Set each factor to zero and solve:
[tex]\[d = 0 \quad \text{or} \quad 3.43d - 6 = 0\][/tex]
[tex]\[d = 0 \quad \text{or} \quad 3.43d = 6\][/tex]
[tex]\[d = \frac{6}{3.43} \approx 1.74927113702624\][/tex]
So, the solutions for [tex]\(d\)[/tex] are approximately:
[tex]\[d = 0 \quad \text{or} \quad d \approx 1.749\][/tex]
### Summary of Solutions
[tex]\[
\begin{aligned}
a & = -50 \\
b & \approx -2.312 \\
c & = 0 \quad \text{or} \quad 2 \\
d & = 0 \quad \text{or} \quad 1.749
\end{aligned}
\][/tex]
These are the solutions for each equation based on the given conditions.
