The
<u>correct answers</u> are:
The <span><u>first system</u> </span>matches with (-4, 5);
The <u>second system</u> matches with (2, 2);
The <u>third sys</u><span><u>tem</u> </span>matches with (-1, -3);
The <u>fourth system</u> matches with (3, 4);
The <u>fifth system</u> matches with (2, -1); and
<span>The <u>sixth system</u> matches with (-5, 6).</span>
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Explanation:</span>
The <u>sixth system</u> is the easiest one to find. We are given the value of x and the value of y in the equations: x=-5 and y=6.
For all other systems, we must solve the system. For the <u>first system</u>:
Since we have the y-variable isolated in the second equation, we will use substitution. We substitute this value in place of y in the first equation:
2x-y=-13
2x-(x+9)=-13
Distributing the subtraction sign,
2x-x-9=-13
Combining like terms:
x-9=-13
Add 9 to each side:
x-9+9=-13+9
x=-4
Substitute this into the second equation:
y=x+9
y=-4+9
y=5
<u>The coordinates (-4, 5) represent the solution point.</u>
For the <u>second system</u>:
We can make the coefficients of x the same and use elimination. To do this, we will multiply the first equation by 2:
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Since the coefficients of x are now the same, we can cancel it. They are both positive, so we subtract:
Divide both sides by 5:
5y/5=10/5
y=2
Substitute this into the second equation
6x-y=10
6x-2=10
Add 2 to each side:
6x-2+2=10+2
6x=12
Divide both sides by 6:
6x/6 = 12/6
x=2
<span><u>The coordinates (2, 2) represent the solution to this system.</u></span>
<span>For the <u>third system</u>:</span>
We can make the coefficients of x the same by multiplying the first equation by 3 and the second by 4:
Since the coefficients of x are the same, we can cancel them. Since they are both positive, we will subtract:
Divide both sides by -17:
-17y/-17 = 51/-17
y=-3
Substitute this into the first equation:
4x-3y=5
4x-3(-3)=5
4x--9=5
4x+9=5
Subtract 9 from each side:
4x+9-9=5-9
4x=-4
Divide each side by 4:
4x/4 = -4/4
x=-1
<u>The coordinates (-1, -3) represent the solution of the third system..</u>
For the <u>fourth system</u>:
Since the coordinates of x are the same and both are positive, we can cancel x by subtracting:
Divide both sides by 2:
2y/2 = 8/2
y=4
Substitute this into the first equation
x+y=7
x+4=7
Subtract 4 from each side:
x+4-4=7-4
x=3
<u>The coordinates (3, 4) represent the solution to the fourth system.</u>
For the <u>fifth system</u>:
Since y is isolated in each equation, we can set them equation to each other:
3x-7=2x-5
Subtract 2x from each side:
3x-7-2x=2x-5-2x
x-7=-5
Add 7 to each side:
x-7+7=-5+7
x=2
Substitute this into the first equation:
y=3x-7
y=3(2)-7
y=6-7
y=-1
<span><u>The coordinates (2, -1) represent the solution to the fifth system.</u></span>
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