Identify any two equations that form a system with one solution (different slopes)

Question

Identify any two equations that form a system with one solution (different slopes)
? I Hint: Make sure all equations are in Y=mx + b format first. J
1. y = 57 -4
2. y = -2x + 3
3. y = 3x +1
4. 2y = 6x + 2
5. у - 5х = 2
6. у -4х +1 = 0

GPT-4o mini · Answer

To identify two equations that form a system with one solution (meaning the lines intersect at exactly one point), we need to ensure they have different slopes.

First, let's rewrite all equations in the format $y = m x + b$ :

$y = 57 - 4$ simplifies to $y = 53$ . (This is a horizontal line with slope = 0.)
$y = - 2 x + 3$ is already in the correct form. (Slope = -2.)
$y = 3 x + 1$ is already in the correct form. (Slope = 3.)
Starting from $2 y = 6 x + 2$ , divide by 2: $y = 3 x + 1$ (This has a slope of 3).
From $y - 5 x = 2$ , we rearrange to get $y = 5 x + 2$ (Slope = 5).
From $y - 4 x + 1 = 0$ , rearranging gives $y = 4 x - 1$ (Slope = 4).

Now we have the slopes for each equation:

$y = 53$ (slope = 0)
$y = - 2 x + 3$ (slope = -2)
$y = 3 x + 1$ (slope = 3)
$y = 3 x + 1$ (slope = 3)
$y = 5 x + 2$ (slope = 5)
$y = 4 x - 1$ (slope = 4)

For the system to have one solution, we need pairs of equations with different slopes.

Let's select:

Equation 2: $y = - 2 x + 3$ (slope = -2)
Equation 5: $y = 5 x + 2$ (slope = 5)

These two equations have different slopes and will intersect at exactly one point, thus forming a system with one solution.

So, the two equations are:

$y = - 2 x + 3$
$y = 5 x + 2$