Question

consider this system of equations:
x - 6y = 4
3x - 18y = 4
change one number to make a new system with one solution.
equation 1
equation 2
reset

Explanation:

Step1: Analyze the original system

The first equation is \(x - 6y = 4\). The second equation is \(3x - 18y = 4\). Notice that if we multiply the first equation by 3, we get \(3x - 18y = 12\). But the second equation has \(3x - 18y = 4\), which means the two lines are parallel (same slope, different y - intercepts) and the system has no solution.

Step2: Modify a number to get one solution

To get a system with one solution, we need the two equations to represent lines with different slopes. Let's modify the coefficient of \(y\) in the second equation. Let's change the \(- 18y\) in the second equation to \(- 17y\). So the new system is:
Equation 1: \(x - 6y = 4\)
Equation 2: \(3x - 17y = 4\)

Now, the slope of the first line is \(\frac{1}{6}\) (from \(y=\frac{1}{6}x-\frac{4}{6}\)) and the slope of the second line is \(\frac{3}{17}\) (from \(y = \frac{3}{17}x-\frac{4}{17}\)). Since \(\frac{1}{6}
eq\frac{3}{17}\), the two lines will intersect at one point, so the system has one solution. (There are other possible changes, for example, we could also change the constant term in the second equation. If we change the 4 in the second equation to 12, the two equations will be the same and have infinitely many solutions, which is not what we want. So changing a coefficient of \(x\) or \(y\) to make the slopes different is a good approach.)

Answer:

Equation 1: \(x - 6y = 4\)
Equation 2: \(3x - 17y = 4\) (Note: There are multiple correct answers. Another example: Equation 1: \(x - 5y = 4\), Equation 2: \(3x - 18y = 4\) would also work as the slopes \(\frac{1}{5}\) and \(\frac{3}{18}=\frac{1}{6}\) are different.)