Let's carefully examine the given equation step-by-step:
### Step 1: Write Down the Equation
Given equation:
[tex]\[
(4x + 1)(3x - 7) - 2x(6x + 5) = 0
\][/tex]
### Step 2: Expand the Equation
First, we need to expand each term using the distributive property (i.e., multiplying through):
For \((4x + 1)(3x - 7)\):
[tex]\[
(4x + 1)(3x - 7) = 4x \cdot 3x + 4x \cdot (-7) + 1 \cdot 3x + 1 \cdot (-7)
\][/tex]
[tex]\[
= 12x^2 - 28x + 3x - 7
\][/tex]
[tex]\[
= 12x^2 - 25x - 7
\][/tex]
For \(-2x(6x + 5)\):
[tex]\[
-2x(6x + 5) = -2x \cdot 6x - 2x \cdot 5
\][/tex]
[tex]\[
= -12x^2 - 10x
\][/tex]
### Step 3: Combine and Simplify
Combine the expanded results:
[tex]\[
12x^2 - 25x - 7 - 12x^2 - 10x = 0
\][/tex]
Combine like terms:
[tex]\[
12x^2 - 12x^2 - 25x - 10x - 7 = 0
\][/tex]
[tex]\[
-35x - 7 = 0
\][/tex]
### Step 4: Determine Linearity
An equation is linear if its highest degree of the variable \(x\) is 1. Since the reduced form of our equation is:
[tex]\[
-35x - 7 = 0
\][/tex]
It is indeed linear because the highest degree of \(x\) in the equation is 1.
### Step 5: Solve for \(x\)
Solve the linear equation:
[tex]\[
-35x - 7 = 0
\][/tex]
Add 7 to both sides:
[tex]\[
-35x = 7
\][/tex]
Divide both sides by -35:
[tex]\[
x = \frac{7}{-35}
\][/tex]
[tex]\[
x = -\frac{1}{5}
\][/tex]
### Conclusion
The given equation is linear, and the solution for \(x\) is:
[tex]\[
x = -\frac{1}{5}
\][/tex]
