Answer :
Sure! Let's find the missing entries in the given table for the equation \(9x - 6y = 18\). We will determine the \(y\) values corresponding to the given \(x\) values and the \(x\) value corresponding to the given \(y\) value.
### Step 1: Solve for \(y\) when \(x = 1\)
Substitute \(x = 1\) into the equation \(9x - 6y = 18\):
[tex]\[ 9(1) - 6y = 18 \implies 9 - 6y = 18 \][/tex]
[tex]\[ -6y = 18 - 9 \implies -6y = 9 \][/tex]
[tex]\[ y = \frac{9}{-6} \implies y = -\frac{3}{2} \implies y = -1.5 \][/tex]
So, when \(x = 1\), \(y = -1.5\).
### Step 2: Solve for \(y\) when \(x = 0\)
Substitute \(x = 0\) into the equation \(9x - 6y = 18\):
[tex]\[ 9(0) - 6y = 18 \implies 0 - 6y = 18 \][/tex]
[tex]\[ -6y = 18 \][/tex]
[tex]\[ y = \frac{18}{-6} \implies y = -3 \][/tex]
So, when \(x = 0\), \(y = -3\).
### Step 3: Solve for \(y\) when \(x = 3\)
Substitute \(x = 3\) into the equation \(9x - 6y = 18\):
[tex]\[ 9(3) - 6y = 18 \implies 27 - 6y = 18 \][/tex]
[tex]\[ -6y = 18 - 27 \implies -6y = -9 \][/tex]
[tex]\[ y = \frac{-9}{-6} \implies y = \frac{3}{2} \implies y = 1.5 \][/tex]
So, when \(x = 3\), \(y = 1.5\).
### Step 4: Solve for \(x\) when \(y = 0\)
Substitute \(y = 0\) into the equation \(9x - 6y = 18\):
[tex]\[ 9x - 6(0) = 18 \implies 9x - 0 = 18 \][/tex]
[tex]\[ 9x = 18 \][/tex]
[tex]\[ x = \frac{18}{9} \implies x = 2 \][/tex]
So, when \(y = 0\), \(x = 2\).
### Summary
The completed table with the missing entries is:
[tex]\[ \begin{tabular}{|l|l|l|l|l|} \hline [tex]$x$[/tex] & 1 & 0 & 3 & 2 \\
\hline [tex]$y$[/tex] & -1.5 & -3 & 1.5 & 0 \\
\hline
\end{tabular}
\][/tex]
Thus, the values of the missing entries satisfy the given equation.
### Step 1: Solve for \(y\) when \(x = 1\)
Substitute \(x = 1\) into the equation \(9x - 6y = 18\):
[tex]\[ 9(1) - 6y = 18 \implies 9 - 6y = 18 \][/tex]
[tex]\[ -6y = 18 - 9 \implies -6y = 9 \][/tex]
[tex]\[ y = \frac{9}{-6} \implies y = -\frac{3}{2} \implies y = -1.5 \][/tex]
So, when \(x = 1\), \(y = -1.5\).
### Step 2: Solve for \(y\) when \(x = 0\)
Substitute \(x = 0\) into the equation \(9x - 6y = 18\):
[tex]\[ 9(0) - 6y = 18 \implies 0 - 6y = 18 \][/tex]
[tex]\[ -6y = 18 \][/tex]
[tex]\[ y = \frac{18}{-6} \implies y = -3 \][/tex]
So, when \(x = 0\), \(y = -3\).
### Step 3: Solve for \(y\) when \(x = 3\)
Substitute \(x = 3\) into the equation \(9x - 6y = 18\):
[tex]\[ 9(3) - 6y = 18 \implies 27 - 6y = 18 \][/tex]
[tex]\[ -6y = 18 - 27 \implies -6y = -9 \][/tex]
[tex]\[ y = \frac{-9}{-6} \implies y = \frac{3}{2} \implies y = 1.5 \][/tex]
So, when \(x = 3\), \(y = 1.5\).
### Step 4: Solve for \(x\) when \(y = 0\)
Substitute \(y = 0\) into the equation \(9x - 6y = 18\):
[tex]\[ 9x - 6(0) = 18 \implies 9x - 0 = 18 \][/tex]
[tex]\[ 9x = 18 \][/tex]
[tex]\[ x = \frac{18}{9} \implies x = 2 \][/tex]
So, when \(y = 0\), \(x = 2\).
### Summary
The completed table with the missing entries is:
[tex]\[ \begin{tabular}{|l|l|l|l|l|} \hline [tex]$x$[/tex] & 1 & 0 & 3 & 2 \\
\hline [tex]$y$[/tex] & -1.5 & -3 & 1.5 & 0 \\
\hline
\end{tabular}
\][/tex]
Thus, the values of the missing entries satisfy the given equation.
