$$ 3 x_{1}+2 x_{2}=6 $$ $$ 4 x_{1}+4 x_{2}=8 $$ find $$ x_{1} $$ and $$ x_{2} $$

Question

UpStudy AI · Accepted Answer

To solve the system of linear equations:

$$
\begin{cases}
3x_{1} + 2x_{2} = 6 \
4x_{1} + 4x_{2} = 8 
\end{cases}
$$

**Step 1: Simplify the Second Equation**

Notice that the second equation can be simplified by dividing all terms by 4:

$$
4x_{1} + 4x_{2} = 8 \quad \Rightarrow \quad x_{1} + x_{2} = 2
$$

**Step 2: Express $$x_{1}$$ in Terms of $$x_{2}$$**

From the simplified second equation:

$$
x_{1} = 2 - x_{2}
$$

**Step 3: Substitute $$x_{1}$$ into the First Equation**

Substitute $$x_{1} = 2 - x_{2}$$ into the first equation:

$$
3(2 - x_{2}) + 2x_{2} = 6 \
6 - 3x_{2} + 2x_{2} = 6 \
6 - x_{2} = 6
$$

**Step 4: Solve for $$x_{2}$$**

$$
- x_{2} = 6 - 6 \
- x_{2} = 0 \
x_{2} = 0
$$

**Step 5: Find $$x_{1}$$**

Using $$x_{1} = 2 - x_{2}$$:

$$
x_{1} = 2 - 0 = 2
$$

**Solution:**

$$
x_{1} = 2 \quad \text{and} \quad x_{2} = 0
$$
$$ x_{1} = 2 $$ and $$ x_{2} = 0 $$.

\( 3 x_{1}+2 x_{2}=6 \) \( 4 x_{1}+4 x_{2}=8 \) find \( x_{1} \) and \( x_{2} \)