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Answer :
We are given the system of equations
[tex]$$
\begin{aligned}
x - 2y &= 8 \quad \text{(1)} \\
3x - 6y &= 6 \quad \text{(2)}
\end{aligned}
$$[/tex]
We will solve the system algebraically and graphically.
──────────────────────────────
Step 1: Algebraic Method
Notice that if we multiply Equation (1) by 3, we obtain:
[tex]$$
3(x - 2y) = 3(8) \quad \Longrightarrow \quad 3x - 6y = 24.
$$[/tex]
Now, compare this with Equation (2), which is:
[tex]$$
3x - 6y = 6.
$$[/tex]
Since
[tex]$$
24 \neq 6,
$$[/tex]
there is a contradiction. This tells us that the system is inconsistent, meaning there is no solution.
──────────────────────────────
Step 2: Graphical Method
We can write each equation in slope-intercept form ([tex]$y = mx + b$[/tex]).
For Equation (1):
[tex]$$
x - 2y = 8 \quad \Longrightarrow \quad -2y = -x + 8 \quad \Longrightarrow \quad y = \frac{1}{2}x - 4.
$$[/tex]
For Equation (2):
[tex]$$
3x - 6y = 6 \quad \Longrightarrow \quad -6y = -3x + 6 \quad \Longrightarrow \quad y = \frac{1}{2}x - 1.
$$[/tex]
Both equations have the same slope, [tex]$m = \frac{1}{2}$[/tex], which means the lines are parallel. However, their [tex]$y$[/tex]-intercepts are different ([tex]$-4$[/tex] for Equation (1) and [tex]$-1$[/tex] for Equation (2)). Since parallel lines with different intercepts never intersect, there is no point of intersection.
──────────────────────────────
Conclusion
The system of equations has no solution.
An important intermediate check shows that multiplying the right-hand side of Equation (1) by 3 gives
[tex]$$
3 \times 8 = 24,
$$[/tex]
while the corresponding right-hand side in Equation (2) is 6. The difference, a contradiction, is
[tex]$$
24 - 6 = 18.
$$[/tex]
This confirms that no solution exists for the system.
[tex]$$
\begin{aligned}
x - 2y &= 8 \quad \text{(1)} \\
3x - 6y &= 6 \quad \text{(2)}
\end{aligned}
$$[/tex]
We will solve the system algebraically and graphically.
──────────────────────────────
Step 1: Algebraic Method
Notice that if we multiply Equation (1) by 3, we obtain:
[tex]$$
3(x - 2y) = 3(8) \quad \Longrightarrow \quad 3x - 6y = 24.
$$[/tex]
Now, compare this with Equation (2), which is:
[tex]$$
3x - 6y = 6.
$$[/tex]
Since
[tex]$$
24 \neq 6,
$$[/tex]
there is a contradiction. This tells us that the system is inconsistent, meaning there is no solution.
──────────────────────────────
Step 2: Graphical Method
We can write each equation in slope-intercept form ([tex]$y = mx + b$[/tex]).
For Equation (1):
[tex]$$
x - 2y = 8 \quad \Longrightarrow \quad -2y = -x + 8 \quad \Longrightarrow \quad y = \frac{1}{2}x - 4.
$$[/tex]
For Equation (2):
[tex]$$
3x - 6y = 6 \quad \Longrightarrow \quad -6y = -3x + 6 \quad \Longrightarrow \quad y = \frac{1}{2}x - 1.
$$[/tex]
Both equations have the same slope, [tex]$m = \frac{1}{2}$[/tex], which means the lines are parallel. However, their [tex]$y$[/tex]-intercepts are different ([tex]$-4$[/tex] for Equation (1) and [tex]$-1$[/tex] for Equation (2)). Since parallel lines with different intercepts never intersect, there is no point of intersection.
──────────────────────────────
Conclusion
The system of equations has no solution.
An important intermediate check shows that multiplying the right-hand side of Equation (1) by 3 gives
[tex]$$
3 \times 8 = 24,
$$[/tex]
while the corresponding right-hand side in Equation (2) is 6. The difference, a contradiction, is
[tex]$$
24 - 6 = 18.
$$[/tex]
This confirms that no solution exists for the system.
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