Certainly! Let's solve the given system of equations step by step:
The given system of equations is:
1. [tex]\(2x - 3y - 5 = 2(y - 3)\)[/tex]
2. [tex]\(3x - y - 2 = 0\)[/tex]
Let's start by simplifying Equation 1.
Step 1: Simplify Equation 1
Given:
[tex]\[2x - 3y - 5 = 2(y - 3)\][/tex]
Distribute the 2 on the right side:
[tex]\[2x - 3y - 5 = 2y - 6\][/tex]
Now, combine like terms:
[tex]\[2x - 3y - 2y = -6 + 5\][/tex]
[tex]\[2x - 5y = -1 \][/tex]
This simplifies Equation 1 to:
[tex]\[2x - 5y = -1 \quad \text{(Equation 3)}\][/tex]
So now, we have the simplified system of equations:
3. [tex]\(2x - 5y = -1\)[/tex]
4. [tex]\(3x - y - 2 = 0 \quad \text{(Equation 2)}\)[/tex]
Step 2: Solve the system of equations
We will solve Equation 4 for y and substitute it into Equation 3.
Solve Equation 4 for y:
[tex]\[3x - y - 2 = 0\][/tex]
[tex]\[ - y = - 3x + 2 \][/tex]
[tex]\[ y = 3x - 2 \][/tex]
Step 3: Substitute y into Equation 3
Substitute [tex]\( y = 3x - 2 \)[/tex] into Equation 3:
[tex]\[2x - 5(3x - 2) = -1 \][/tex]
Distribute the [tex]\(-5\)[/tex]:
[tex]\[2x - 15x + 10 = -1\][/tex]
Combine like terms:
[tex]\[-13x + 10 = -1\][/tex]
[tex]\[-13x = -1 - 10\][/tex]
[tex]\[-13x = -11\][/tex]
Solve for [tex]\(x\)[/tex]:
[tex]\[x = \frac{11}{13} \][/tex]
Step 4: Substitute [tex]\(x\)[/tex] back into the equation for [tex]\(y\)[/tex]
Now substitute [tex]\( x = \frac{11}{13}\)[/tex] into the equation [tex]\( y = 3x - 2\)[/tex]:
[tex]\[ y = 3\left(\frac{11}{13}\right) - 2 \][/tex]
[tex]\[ y = \frac{33}{13} - 2 \][/tex]
[tex]\[ y = \frac{33}{13} - \frac{26}{13} \][/tex]
[tex]\[ y = \frac{7}{13} \][/tex]
So the solution to the system of equations is:
[tex]\[ x = \frac{11}{13}\][/tex]
[tex]\[ y = \frac{7}{13}\][/tex]
