To determine the value of [tex]\( r \)[/tex] such that the given system of equations has no solution, we need to follow several mathematical steps focusing on the properties of a system of linear equations. The given system of equations is:
[tex]\[
\begin{aligned}
48x - 72y &= 30y + 24 \\
ry &= \frac{1}{6} - 16x
\end{aligned}
\][/tex]
### Step 1: Simplify the First Equation
We start by simplifying the first equation:
[tex]\[
48x - 72y = 30y + 24
\][/tex]
Move all terms to one side:
[tex]\[
48x - 72y - 30y = 24
\][/tex]
Combine like terms:
[tex]\[
48x - 102y = 24
\][/tex]
This can be rearranged:
[tex]\[
48x - 102y - 24 = 0 \quad \text{(Equation 1)}
\][/tex]
### Step 2: Rearrange and Simplify the Second Equation
Next, we look at the second equation:
[tex]\[
ry = \frac{1}{6} - 16x
\][/tex]
Rearrange this to put it in a similar form:
[tex]\[
ry + 16x = \frac{1}{6} \quad \text{(Equation 2)}
\][/tex]
### Step 3: Represent Equations in Matrix Form
To determine when the system has no solution (is inconsistent), we express the equations in matrix form and examine the coefficient matrix. The matrix form is:
[tex]\[
\begin{pmatrix}
48 & -102 \\
16 & r
\end{pmatrix}
\begin{pmatrix}
x \\
y
\end{pmatrix}
=
\begin{pmatrix}
24 \\
\frac{1}{6}
\end{pmatrix}
\][/tex]
### Step 4: Find the Determinant of the Coefficient Matrix
The determinant of the coefficient matrix must be zero for the system to have no solutions (indicating parallel lines or intersecting the axes in such a way that they never meet). The coefficient matrix is:
[tex]\[
\begin{pmatrix}
48 & -102 \\
16 & r
\end{pmatrix}
\][/tex]
The determinant ([tex]\( \Delta \)[/tex]) of this [tex]\(2 \times 2\)[/tex] matrix is given by:
[tex]\[
\Delta = 48r - (16 \times -102)
\][/tex]
Simplify the expression:
[tex]\[
\Delta = 48r + 1632
\][/tex]
### Step 5: Set the Determinant to Zero
For the system to have no solution, the determinant must be zero:
[tex]\[
48r + 1632 = 0
\][/tex]
Solve for [tex]\( r \)[/tex]:
[tex]\[
48r = -1632
\][/tex]
[tex]\[
r = -\frac{1632}{48}
\][/tex]
Simplify the fraction:
[tex]\[
r = -34
\][/tex]
### Conclusion
The value of [tex]\( r \)[/tex] that makes the given system of equations have no solution is:
[tex]\[
\boxed{-34}
\][/tex]
