Math  /  Algebra

QuestionWhat is the solution to this system of equations? {3x2y=86x+4y=16\left\{\begin{array}{r} 3 x-2 y=-8 \\ -6 x+4 y=16 \end{array}\right. A. Infinite Solutions B. No Solution C. (0,3)(0,3)

Studdy Solution

STEP 1

What is this asking? We're trying to find the mystery values of xx and yy that make *both* of these equations true at the same time! Watch out! Don't get tricked by equations that look different but are secretly the same!

STEP 2

1. Check for sneaky identical equations
2. Solve for the variables

STEP 3

Let's multiply both sides of the first equation, 3x2y=83x - 2y = -8, by **2**.
Why? Because we're on the hunt for identical equations in disguise!
This gives us 2(3x2y)=2(8)2 \cdot (3x - 2y) = 2 \cdot (-8), which simplifies to 6x4y=166x - 4y = -16.

STEP 4

Now, look at the second equation: 6x+4y=16-6x + 4y = 16.
If we multiply *both* sides of our new equation from the previous sub-step by 1-1, we get (1)(6x4y)=(1)(16)(-1) \cdot (6x - 4y) = (-1) \cdot (-16), which is 6x+4y=16-6x + 4y = 16.
Whoa! This is *exactly* the same as the second equation!

STEP 5

This means the two equations are actually the *same* line, just dressed up differently.
So, any solution that works for one equation will *automatically* work for the other.

STEP 6

Since the equations are the same, we can just pick one to solve.
Let's use 3x2y=83x - 2y = -8.
We want to get yy by itself.
First, let's subtract 3x3x from both sides: 3x2y3x=83x3x - 2y - 3x = -8 - 3x, which simplifies to 2y=83x-2y = -8 - 3x.

STEP 7

Now, divide both sides by 2-2 to isolate yy: 2y2=83x2\frac{-2y}{-2} = \frac{-8 - 3x}{-2}.
This gives us y=82+3x2y = \frac{-8}{-2} + \frac{-3x}{-2}, which simplifies to y=4+32xy = 4 + \frac{3}{2}x.

STEP 8

This equation tells us that for *any* value of xx we choose, we can find a corresponding value of yy that satisfies both equations.
That means there are *infinitely many* solutions!

STEP 9

The answer is **A.
Infinite Solutions**.

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