Question

How many solutions does the system have? You can use the interactive graph below to find the answer. \begin{cases} 4x-2y=8 \\\\ 2x+y=2 \end{cases} ⎩ ⎪ ⎪ ⎨ ⎪ ⎪ ⎧ ​ 4x−2y=8 2x+y=2 ​ Choose 1 answer: Choose 1 answer: (Choice A) A Exactly one solution (Choice B) B No solutions (Choice C) C Infinitely many solutions

Answer 1

Answer: No Solution

Let's bring both equations to slope-intercept form. Then we can think about the slopes and the y-intercepts of the lines represented by each equation.

The slope-intercept form of the first equation 2y=4x+62y=4x+62, y, equals, 4, x, plus, 6 is y=2x+3y=2x+3y, equals, 2, x, plus, 3. The second equation y = 2x+6y=2x+6y, equals, 2, x, plus, 6 is already in slope-intercept form.

Hint #22 / 3

The first equation is y = 2x+3y=2x+3y, equals, 2, x, plus, 3, so the slope of its line is 222 and the yyy-intercept is (0,3)(0,3)left parenthesis, 0, comma, 3, right parenthesis.

The second equation is y = 2x+6y=2x+6y, equals, 2, x, plus, 6, so the slope of its line is 222 and the yyy-intercept is (0,6)(0,6)left parenthesis, 0, comma, 6, right parenthesis.

Since both lines have the same slopes but different yyy-intercepts, they are distinct parallel lines.

Hint #33 / 3

Since distinct parallel lines don't intersect, we conclude that the system has no solutions.

Answer 2

`\Huge\boxed{\textsf{A. Exactly one solution}}`

We have the following system:

`\begin{cases}4x-2y&=8\\2x+y&=2\end{cases}`

Here, a simple solution to find the answer is to graph the two lines and see how many times they intersect.

I've attached a graph, with
`4x-2y=8` in red and the other equation in blue.

See that the lines only intersect once, at
`(1.5,-1)`. This means the system only has one solution.