Let's solve the given system of equations to determine if there are any solutions and, if so, what they are:
1. Rewrite the equations in slope-intercept form (i.e., [tex]\( y = mx + b \)[/tex]):
- For the first equation [tex]\( 3y + 3x = 2 \)[/tex]:
[tex]\[
3y + 3x = 2 \implies 3y = 2 - 3x \implies y = \frac{2 - 3x}{3}
\][/tex]
- For the second equation [tex]\( y + x = 8 \)[/tex]:
[tex]\[
y + x = 8 \implies y = 8 - x
\][/tex]
2. Set the right-hand sides of the equations equal to each other to find the intersection point:
[tex]\[
\frac{2 - 3x}{3} = 8 - x
\][/tex]
3. Clear the fraction by multiplying both sides by 3:
[tex]\[
2 - 3x = 3(8 - x)
\][/tex]
[tex]\[
2 - 3x = 24 - 3x
\][/tex]
4. Simplify the equation:
Notice that both sides have a [tex]\(-3x\)[/tex]:
[tex]\[
2 - 3x + 3x = 24 - 3x + 3x \implies 2 = 24
\][/tex]
5. Analyze the resulting equation:
The simplified equation is [tex]\( 2 = 24 \)[/tex], which is a contradiction because 2 does not equal 24.
This contradiction means that there is no [tex]\(x\)[/tex] and [tex]\(y\)[/tex] pair that satisfies both equations simultaneously. Therefore, the system of equations represents parallel lines that do not intersect at any point.
Thus, the correct answer is:
D. No solution
