Math  /  Algebra

Question8x6y=572x+54y=45\begin{aligned} 8 x-6 y & =5 \\ -72 x+54 y & =-45\end{aligned} system or - equations 55 \quad (1) \qquad (2)
4. The solution to this system is c. There is no solution. which solution. solutions. \square the 0_{0} \qquad Choices \qquad nd fill \qquad \qquad present in 2^{2} Which of the statem ments ss below is \qquad
The graphs intersect at one point hever intersect. \square so the solution is unique.

Studdy Solution

STEP 1

What is this asking? We've got two equations with *x* and *y*, and we need to figure out if they have a solution, meaning if there's a value for *x* and a value for *y* that make both equations true at the same time! Watch out! Sometimes equations that look different can actually be the same equation in disguise, so we need to be careful!

STEP 2

1. Simplify the equations
2. Compare the equations

STEP 3

Let's look at our first equation: 8x6y=58x - 6y = 5.
This looks pretty simple already!
We can't really simplify it further.

STEP 4

Now, let's check out the second equation: 72x+54y=45-72x + 54y = -45.
Hmm, those numbers look kinda big.
I bet we can simplify this!
Let's **divide both sides** by 9-9.
Remember, we can do this as long as we do it to both sides of the equation!
Why 9-9?
Because it's the **greatest common factor** of -72, 54, and -45.

STEP 5

Dividing both sides by 9-9 gives us: 72x9+54y9=459 \frac{-72x}{-9} + \frac{54y}{-9} = \frac{-45}{-9} 8x6y=5 8x - 6y = 5 Whoa! This is the **exact same** as our first equation!

STEP 6

Since both equations simplify to the same thing, they're actually the same line!
If we were to graph them, they'd lie right on top of each other.

STEP 7

This means there are infinitely many points where the lines "intersect" because they're always intersecting!
Every point on the line is a solution!

STEP 8

There are infinitely many solutions because the two equations represent the same line!

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