Change one number in the following system of equations to create a new system with one solution: x - 6y = 4 ; 3x - 18y = 4

Steps to Solve

1. Analyze the given system

The system of equations is: $3x + 4y = 7$ $6x + 8y = 10$

Notice that the coefficients of $x$ and $y$ in the second equation are exactly twice the coefficients in the first equation. This is why the lines are parallel and there is no solution.

2. Understand the condition for a unique solution

For the system to have one solution, the ratio of the coefficients of $x$ and $y$ must not be equal. That is, for the system $ax + by = c$ $dx + ey = f$ we must have $\frac{a}{d} \neq \frac{b}{e}$.

3. Consider modifying the constant term

Changing the constant term in the second equation would only shift the line, and it would remain parallel to the first equation. So, changing the constant term will not result in one solution. We can see this as follows: changing $10$ to some other number $k$ results in the equations $3x + 4y = 7$ $6x + 8y = k$ which means that $2(3x + 4y) = 2(7) = 14 = k$. So only when $k = 14$ will there be infinitely many solutions. If $k$ is any other number there will be no solution.

4. Modify the coefficient of $x$ in the second equation

Let's change the coefficient of $x$ in the second equation from 6 to some other number $k$. Then the equations become:

$3x + 4y = 7$ $kx + 8y = 10$

For a unique solution, we require $\frac{3}{k} \neq \frac{4}{8}$, which simplifies to $\frac{3}{k} \neq \frac{1}{2}$. Thus, $k \neq 6$. Any value other than 6 will work. For example, let $k = 1$. Then the second equation becomes $x + 8y = 10$. Then the new system has one solution.

5. Solve the modified system (example)

Let's say we changed 6 to 1 as described above, giving us the system:

$3x + 4y = 7$ $x + 8y = 10$

Multiply the second equation by 3: $3x + 24y = 30$. Subtract the first equation from this: $(3x + 24y) - (3x + 4y) = 30 - 7$, which simplifies to $20y = 23$, so $y = \frac{23}{20}$. Substitute this value of $y$ into the second equation: $x + 8(\frac{23}{20}) = 10$, so $x + \frac{46}{5} = 10$, and $x = 10 - \frac{46}{5} = \frac{50 - 46}{5} = \frac{4}{5}$. Therefore, the unique solution is $x = \frac{4}{5}$ and $y = \frac{23}{20}$.

6. Determine the change made

We changed the coefficient of $x$ from 6 to 1. The question asks "What number can you change to make the new system have exactly one solution?" Therefore the number that we changed is 6, and it was changed to 1.