Change one number to make a new system with one solution.

Understand the Problem

The question is asking to modify one number in a given system of equations so that the modified system has exactly one solution, which typically occurs when the equations are neither parallel nor identical.

Answer

Change Equation 1 to $x - 6y = 5$.

Steps to Solve

Identify the Current System's Characteristics

The given system is:

$x - 6y = 4$ (Equation 1)
$3x - 18y = 4$ (Equation 2)

Equation 2 can be simplified (divided by 3) to: $$ x - 6y = \frac{4}{3} $$

Both equations simplify to the same line when ignoring constants, suggesting they are identical lines.

Choose a Modification

To achieve a system with exactly one solution, we need to alter one equation so that it has a different slope or y-intercept.

Modify the Constant in Equation 1

We could change the constant in Equation 1 by altering the right side:

New Equation 1: $x - 6y = 5$

Now the system is:

$x - 6y = 5$
$3x - 18y = 4$
Verify the New System

Substituting $y$ from the first equation gives: $$ y = \frac{x - 5}{6} $$

Substituting into the second equation: $$ 3x - 18\left(\frac{x - 5}{6}\right) = 4 $$

This yields different equations, ensuring the lines are not parallel.

Change the constant in Equation 1: $x - 6y = 5$.

More Information

By changing the constant of the first equation, the two equations now represent different lines, which guarantees a unique intersection point, hence one solution.

Tips

Changing coefficients instead of constants can result in parallel lines.
Ensuring that the two equations have distinct slopes is key for one unique solution.

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