Certainly! Let's solve the system of linear equations step-by-step.
Given the system of equations:
1. [tex]\(3x - 2y = 12\)[/tex]
2. [tex]\(x = -2y - 4\)[/tex]
We can use substitution to solve this system.
Step 1: Substitute the second equation into the first equation.
From the second equation, [tex]\(x = -2y - 4\)[/tex].
Substitute this expression for [tex]\(x\)[/tex] in the first equation:
[tex]\[3(-2y - 4) - 2y = 12\][/tex]
Step 2: Simplify the equation.
First, distribute the 3 in the equation:
[tex]\[3 \cdot (-2y) + 3 \cdot (-4) - 2y = 12\][/tex]
This simplifies to:
[tex]\[-6y - 12 - 2y = 12\][/tex]
Step 3: Combine like terms.
Combine the terms involving [tex]\(y\)[/tex]:
[tex]\[-6y - 2y - 12 = 12\][/tex]
This simplifies to:
[tex]\[-8y - 12 = 12\][/tex]
Step 4: Isolate the term involving [tex]\(y\)[/tex].
Add 12 to both sides of the equation to isolate the [tex]\(y\)[/tex] term:
[tex]\[-8y - 12 + 12 = 12 + 12\][/tex]
[tex]\[-8y = 24\][/tex]
Step 5: Solve for [tex]\(y\)[/tex].
Divide both sides by -8:
[tex]\[y = \frac{24}{-8}\][/tex]
[tex]\[y = -3\][/tex]
Step 6: Substitute [tex]\(y\)[/tex] back into the second equation to solve for [tex]\(x\)[/tex].
Now that we have [tex]\(y = -3\)[/tex], we can substitute it back into the second equation:
[tex]\[x = -2(-3) - 4\][/tex]
Simplify:
[tex]\[x = 6 - 4\][/tex]
[tex]\[x = 2\][/tex]
Final Solution:
The solution to the system of equations is:
[tex]\[x = 2\][/tex]
[tex]\[y = -3\][/tex]
Thus, the solution as an ordered pair is [tex]\((2, -3)\)[/tex].
