Solve the following system of equations using the graphical method:

[tex]\[
\begin{array}{l}
x + 3y = 2 \\
x + \frac{1}{3}x = 1
\end{array}
\][/tex]

[tex]\[
\begin{array}{l}
3x + 2y = 16 \\
5x - 3y = 14
\end{array}
\][/tex]



Answer :

To solve the system of equations:

[tex]\[ \begin{cases} 3x + 2y = 16 \\ 5x - 3y = 14 \end{cases} \][/tex]

we'll follow a step-by-step algebraic method to find the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex].

### Step 1: Write Down the Equations
We have the two linear equations:
[tex]\[ 3x + 2y = 16 \quad \text{(Equation 1)} \][/tex]
[tex]\[ 5x - 3y = 14 \quad \text{(Equation 2)} \][/tex]

### Step 2: Prepare to Eliminate One Variable
We'll first eliminate [tex]\( y \)[/tex]. To do this, we'll make the coefficients of [tex]\( y \)[/tex] in both equations the same by finding a common multiple. The least common multiple of 2 and 3 is 6. Therefore:

Multiply Equation 1 by 3:
[tex]\[ 3(3x + 2y) = 3(16) \Rightarrow 9x + 6y = 48 \quad \text{(Equation 3)} \][/tex]

Multiply Equation 2 by 2:
[tex]\[ 2(5x - 3y) = 2(14) \Rightarrow 10x - 6y = 28 \quad \text{(Equation 4)} \][/tex]

### Step 3: Add the Equations to Eliminate [tex]\( y \)[/tex]
Now, we'll add Equation 3 and Equation 4:
[tex]\[ (9x + 6y) + (10x - 6y) = 48 + 28 \][/tex]

This simplifies to:
[tex]\[ 9x + 10x = 76 \][/tex]
[tex]\[ 19x = 76 \][/tex]

Solving for [tex]\( x \)[/tex]:
[tex]\[ x = \frac{76}{19} = 4 \][/tex]

### Step 4: Substitute [tex]\( x = 4 \)[/tex] into One of the Original Equations
We'll use [tex]\( x = 4 \)[/tex] in Equation 1 to find [tex]\( y \)[/tex]:
[tex]\[ 3(4) + 2y = 16 \][/tex]
[tex]\[ 12 + 2y = 16 \][/tex]

Solving for [tex]\( y \)[/tex]:
[tex]\[ 2y = 16 - 12 \][/tex]
[tex]\[ 2y = 4 \][/tex]
[tex]\[ y = \frac{4}{2} = 2 \][/tex]

### Step 5: State the Solution
The solution to the system of equations is:
[tex]\[ (x, y) = (4, 2) \][/tex]

This means the values of [tex]\( x \)[/tex] and [tex]\( y \)[/tex] that satisfy both equations are [tex]\( x = 4 \)[/tex] and [tex]\( y = 2 \)[/tex].