Question

One of the two linear equations in a system is given below. This system has no solutions. Which equation could be the second equation in this system?

2x−12y=8
x−6y=4
x−4y=0
2
1
​
x−3y=2
2
1
​
−3y=0

Answer 1

For a system of linear equations to have no solutions, the two equations must be parallel. This means they have the same slope but different y-intercepts.

Let's first express the given equation, \(2x - 12y = 8\), in slope-intercept form (\(y = mx + b\)):

1. Start with the equation:
\[
2x - 12y = 8
\]

2. Isolate \(y\):
\[
-12y = -2x + 8
\]
\[
y = \frac{1}{6}x - \frac{2}{3}
\]

The slope \(m\) of this line is \(\frac{1}{6}\).

Now, let's analyze the options provided to find which equation has the same slope of \(\frac{1}{6}\).

1. **Option: \(x - 6y = 4\)**
\[
-6y = -x + 4
\]
\[
y = \frac{1}{6}x - \frac{2}{3}
\]
(Same slope \(\frac{1}{6}\))

2. **Option: \(x - 4y = 0\)**
\[
-4y = -x
\]
\[
y = \frac{1}{4}x
\]
(Slope \(\frac{1}{4}\), not parallel)

3. **Option: \(x - 3y = 2\)**
\[
-3y = -x + 2
\]
\[
y = \frac{1}{3}x - \frac{2}{3}
\]
(Slope \(\frac{1}{3}\), not parallel)

4. **Option: \(-3y = 0\)**
This equation yields no variable to compare directly, as it's simply \(y = 0\), which is a horizontal line (not parallel).

The only equation that would result in parallel lines, and thus a system with no solutions, is:

**Option:** \(x - 6y = 4\)