To determine which of the given points lies on the graph of the linear equation [tex]\( y = 2x \)[/tex], we'll substitute each point into the equation and verify if the equation holds true.
1. For point [tex]\((2, 1)\)[/tex]:
[tex]\[
\begin{align*}
x &= 2 \\
y &= 1 \\
\end{align*}
Substitute these values into the equation \( y = 2x \):
\[
1 = 2 \cdot 2 \implies 1 = 4
\][/tex]
This is not true. Therefore, point [tex]\((2, 1)\)[/tex] does not lie on the graph.
2. For point [tex]\((2, -1)\)[/tex]:
[tex]\[
\begin{align*}
x &= 2 \\
y &= -1 \\
\end{align*}
Substitute these values into the equation \( y = 2x \):
\[
-1 = 2 \cdot 2 \implies -1 = 4
\][/tex]
This is not true. Therefore, point [tex]\((2, -1)\)[/tex] does not lie on the graph.
3. For point [tex]\(\left(\frac{3}{2}, -3\right)\)[/tex]:
[tex]\[
\begin{align*}
x &= \frac{3}{2} \\
y &= -3 \\
\end{align*}
Substitute these values into the equation \( y = 2x \):
\[
-3 = 2 \cdot \frac{3}{2} \implies -3 = 3
\][/tex]
This is not true. Therefore, point [tex]\(\left(\frac{3}{2}, -3\right)\)[/tex] does not lie on the graph.
4. For point [tex]\(\left(\frac{3}{2}, 3\right)\)[/tex]:
[tex]\[
\begin{align*}
x &= \frac{3}{2} \\
y &= 3 \\
\end{align*}
Substitute these values into the equation \( y = 2x \):
\[
3 = 2 \cdot \frac{3}{2} \implies 3 = 3
\][/tex]
This is true. Therefore, point [tex]\(\left(\frac{3}{2}, 3\right)\)[/tex] lies on the graph.
Thus, the graph of the linear equation [tex]\( y = 2x \)[/tex] passes through the point [tex]\(\left(\frac{3}{2}, 3\right)\)[/tex]. The correct answer is:
(d) [tex]\(\left(\frac{3}{2}, 3\right)\)[/tex].
