Let's consider the given equation \(2y - 4x = 12\).
First, let’s solve this equation for \(y\) in terms of \(x\):
[tex]\[ 2y - 4x = 12 \][/tex]
[tex]\[ 2y = 4x + 12 \][/tex]
[tex]\[ y = 2x + 6 \][/tex]
Hence, the slope of the given equation \(y = 2x + 6\) is \(2\).
Now we need to determine the slopes of the provided equations and see which one forms a system with exactly one solution with the given equation. Remember, for two lines to intersect at one point, they must have different slopes.
Let’s analyze each provided equation:
1. Equation: \(-y - 2x = 6\)
Rewrite in slope-intercept form:
[tex]\[ -y - 2x = 6 \][/tex]
[tex]\[ -y = 2x + 6 \][/tex]
[tex]\[ y = -2x - 6 \][/tex]
The slope here is \(-2\).
2. Equation: \(-y + 2x = 12\)
Rewrite in slope-intercept form:
[tex]\[ -y + 2x = 12 \][/tex]
[tex]\[ -y = -2x + 12 \][/tex]
[tex]\[ y = 2x - 12 \][/tex]
The slope here is \(2\).
3. Equation: \(y = 2x + 6\)
This equation is already in slope-intercept form:
[tex]\[ y = 2x + 6 \][/tex]
The slope here is \(2\).
4. Equation: \(y = 2x + 12\)
This equation is also in slope-intercept form:
[tex]\[ y = 2x + 12 \][/tex]
The slope here is \(2\).
Now, comparing the slopes of the provided equations with the slope of the given equation:
- Equation 1 has slope \(-2\) which is different from \(2\).
- Equation 2 has slope \(2\) which is the same as the given equation.
- Equation 3 has slope \(2\) which is the same as the given equation.
- Equation 4 has slope \(2\) which is the same as the given equation.
Since only Equation 1 (\(-y - 2x = 6\)) has a different slope from the given equation \(2y - 4x = 12\), it will intersect with the given equation at exactly one point, forming a system with one solution.
Therefore, the equation is:
[tex]\[ -y - 2x = 6 \][/tex]
