Question

How many solutions does this system have?

{2x+8y=16
{-3x+6y=30

A. Infinite
B. None
C. One

Either A. Or C.

Answer 1

Step-by-step explanation: The system has one solution, which can be found by solving the equations simultaneously:

-3x + 6y = 30

2x + 8y = 16

Subtracting the second equation from the first:

5x = -16

Divide both sides by 5:

x = -3.2

Now substitute x back into either of the original equations to find y:

-3(-3.2) + 6y = 30

-9.6 + 6y = 30

6y = 39.6

y = 6.4

Therefore, the solution is (-3.2, 6.4).

So the answer is C. One solution.

Answer 2

C. One

Explanation:

You want the number of solutions to the system of equations ...

• 2x +8y = 16
• -3x +6y = 30

Consistent

The signs of the x- and y-terms in the two equations are different. This means the lines have different slopes, so must intersect in one point. (Such equations are called "consistent," because the same (x, y) pair can satisfy both equations.)

There is one solution, choice C.

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Additional comment

We generally like to compare the equations in standard form or slope-intercept form. We can put these equations in standard form by dividing by 2 and -3, respectively:

• x +4y = 8
• x -2y = -10

They are different equations, so must have one solution. (The solution is (x, y) = (-4, 3).)

If they were the same except for the constant on the right, there would be no solutions. If they were the same, there would be infinite solutions.

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