To identify two equations that form a system with one solution (which means they intersect at exactly one point), we need to ensure that the lines represented by the equations have different slopes.
Let's say we start with two equations given in standard form (which is not provided), and we follow your hint to rewrite them into the slope-intercept form, \(y = mx + b\).
Here are two example equations that we can use:
- \(2x + 3y = 6\)
- \(x - 4y = 8\)
Now we need to convert these equations into slope-intercept form.
For the first equation: \[ 2x + 3y = 6 \] Subtract \(2x\) from both sides: \[ 3y = -2x + 6 \] Now divide by 3: \[ y = -\frac{2}{3}x + 2 \] Here, the slope \(m_1 = -\frac{2}{3}\).
For the second equation: \[ x - 4y = 8 \] Subtract \(x\) from both sides: \[ -4y = -x + 8 \] Now divide by -4: \[ y = \frac{1}{4}x - 2 \] Here, the slope \(m_2 = \frac{1}{4}\).
Since the slopes are different (\(-\frac{2}{3} \neq \frac{1}{4}\)), these equations will intersect at one point.
Thus, the system of equations is:
- \(y = -\frac{2}{3}x + 2\)
- \(y = \frac{1}{4}x - 2\)
This system has one solution.