Question

How many solutions does the system of equations have?

y=−2x+12


4y+x−20=0


(1 point)
Responses

No solution
No solution

Infinitely many solutions
Infinitely many solutions

Two solutions
Two solutions

One solution

Answer 1

To determine how many solutions the given system of equations has, we need to analyze the equations:

1. The first equation is \(y = -2x + 12\).
2. The second equation is \(4y + x - 20 = 0\).

We can rewrite the second equation in terms of \(y\):

\[
4y = -x + 20
\]

\[
y = -\frac{1}{4}x + 5
\]

Now, we have the two equations:

1. \(y = -2x + 12\) (a line with a slope of -2)
2. \(y = -\frac{1}{4}x + 5\) (a line with a slope of -\(\frac{1}{4}\))

### Step 1: Compare Slopes
The slopes of the two lines are different (-2 vs -\(\frac{1}{4}\)). When two lines have different slopes, they will intersect at exactly one point.

### Step 2: Finding the Intersection Point
To find the exact intersection, set the equations equal to each other:

\[
-2x + 12 = -\frac{1}{4}x + 5
\]

To eliminate the fraction, multiply the entire equation by 4:

\[
-8x + 48 = -x + 20
\]

Rearranging gives:

\[
-8x + x = 20 - 48
\]

\[
-7x = -28
\]

\[
x = 4
\]

Now, substitute \(x = 4\) back into one of the original equations to find \(y\):

Using \(y = -2x + 12\):

\[
y = -2(4) + 12 = -8 + 12 = 4
\]

Thus, the intersection point is \((4, 4)\).

### Conclusion
Since we found one intersection point, the two equations have **one solution**.

**Answer:** One solution