Certainly! Let's solve the given system of equations step by step.
We start with the system of equations:
[tex]\[
\left[\begin{array}{c}2ab - 1 \\ 4 - b\end{array}\right] = \left[\begin{array}{c}1 \\ 3\end{array}\right]
\][/tex]
This translates to two separate equations:
1. [tex]\(2ab - 1 = 1\)[/tex]
2. [tex]\(4 - b = 3\)[/tex]
Let's solve them one by one.
### Solving Equation 2: [tex]\(4 - b = 3\)[/tex]
Subtract 3 from both sides:
[tex]\[
4 - b - 3 = 0
\][/tex]
This simplifies to:
[tex]\[
b = 1
\][/tex]
So, we have found:
[tex]\[
b = 1
\][/tex]
### Solving Equation 1: [tex]\(2ab - 1 = 1\)[/tex]
Substitute [tex]\(b = 1\)[/tex] into the equation [tex]\(2ab - 1 = 1\)[/tex]:
[tex]\[
2a(1) - 1 = 1
\][/tex]
This simplifies to:
[tex]\[
2a - 1 = 1
\][/tex]
Add 1 to both sides to isolate the term with [tex]\(a\)[/tex]:
[tex]\[
2a - 1 + 1 = 1 + 1
\][/tex]
This further simplifies to:
[tex]\[
2a = 2
\][/tex]
Now, divide both sides by 2:
[tex]\[
a = 1
\][/tex]
So, we have found:
[tex]\[
a = 1
\][/tex]
### Summary
The values of [tex]\(a\)[/tex] and [tex]\(\b\)[/tex] that satisfy both equations are:
[tex]\[
a = 1 \quad \text{and} \quad b = 1
\][/tex]
These are the required values that solve the given system of equations.
