Part 1: Option A : [tex](4,-3)[/tex] is the solution to the equation [tex]3 x+2 y=6[/tex]
Part 2: Option B : [tex]y=-\frac{2}{5} x+1[/tex] is the equation for y.
Part 3: Option D: When [tex]x=-4, y=-9[/tex] and when [tex]x=2, y=3[/tex]
Explanation:
Part 1: The equation is [tex]3 x+2 y=6[/tex]
The solution of the equation can be determined when the ordered pair satisfies the equation [tex]3 x+2 y=6[/tex]
Substituting [tex](4,-3)[/tex] in the equation [tex]3 x+2 y=6[/tex], we have,
[tex]3(4)+2(-3)=6[/tex]
[tex]12-6=6[/tex]
[tex]6=6[/tex]
Thus, the ordered pair [tex](4,-3)[/tex] satisfies the equation [tex]3 x+2 y=6[/tex]
Hence, Option A is the correct answer.
Part 2: The equation is [tex]2 x+5 y=5[/tex]
Let us solve the equation for y.
[tex]5y=-2x+5[/tex]
Dividing both sides by 5, we have,
[tex]y=-\frac{2}{5} x+1[/tex]
Thus, the equation for y is [tex]y=-\frac{2}{5} x+1[/tex]
Hence, Option B is the correct answer.
Part 3: The equation is [tex]3 y-6 x=-3[/tex]
To determine the true statement, we need to substitute the value of x and y in the equation and find out which x and y values results in the equation to be true.
When [tex]x=-4, y=-9[/tex] and when [tex]x=2, y=3[/tex]
First, let us substitute [tex]x=-4, y=-9[/tex] in the equation [tex]3 y-6 x=-3[/tex]
[tex]3(-9)-6(-4)=-3[/tex]
[tex]-27+24=-3[/tex]
[tex]-3=-3[/tex]
Now, let us substitute [tex]x=2, y=3[/tex] in the equation [tex]3 y-6 x=-3[/tex]
[tex]3(3)-6(2)=-3[/tex]
[tex]9-12=-3[/tex]
[tex]-3=-3[/tex]
Thus, the equation is true when [tex]x=-4, y=-9[/tex] and when [tex]x=2, y=3[/tex]
Hence, Option D is the correct answer.
