document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210483>Question 1210483</A>: <I><SPAN STYLE=\"word-break: break-all;\"> graph 3x=y </SPAN>\n" );
document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852858 by </B><A HREF=/tutors/aboutme.mpl?userid=CPhill><B>CPhill(2138)</B></A><img src=\"/images/stars/stars-13.gif\" alt=\"\" >&nbsp;<A HREF=/tutors/aboutme.mpl?userid=CPhill><IMG SRC=http://www.algebra.com/images/aboutme-small.gif BORDER=0 alt=\"About Me\" VALIGN=MIDDLE></A>&nbsp;<IMG SRC=http://www.algebra.com/cgi-bin/show-face.mpl?user=CPhill><BR>You can <A HREF=\"/cgi-bin/embed-solution.mpl?show=1&solution=852858\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> ```python?code_reference&code_event_index=2<BR>\n" );
document.write( "import matplotlib.pyplot as plt<BR>\n" );
document.write( "import numpy as np\r<BR>\n" );
document.write( "\n" );
document.write( "# Define the function<BR>\n" );
document.write( "def line_function(x):<BR>\n" );
document.write( "    return 3 * x\r<BR>\n" );
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document.write( "# Generate x values<BR>\n" );
document.write( "x_values = np.linspace(-5, 5, 400)<BR>\n" );
document.write( "# Generate y values<BR>\n" );
document.write( "y_values = line_function(x_values)\r<BR>\n" );
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document.write( "# Create the plot<BR>\n" );
document.write( "plt.figure(figsize=(8, 6))<BR>\n" );
document.write( "plt.plot(x_values, y_values, label='$y = 3x$')\r<BR>\n" );
document.write( "\n" );
document.write( "# Highlight the origin (y-intercept)<BR>\n" );
document.write( "plt.plot(0, 0, 'ro', label='y-intercept (0, 0)')\r<BR>\n" );
document.write( "\n" );
document.write( "# Add labels and title<BR>\n" );
document.write( "plt.xlabel('$x$')<BR>\n" );
document.write( "plt.ylabel('$y$')<BR>\n" );
document.write( "plt.title('Graph of $3x = y$')\r<BR>\n" );
document.write( "\n" );
document.write( "# Set grid and aspect ratio<BR>\n" );
document.write( "plt.axhline(0, color='black', linewidth=0.5)<BR>\n" );
document.write( "plt.axvline(0, color='black', linewidth=0.5)<BR>\n" );
document.write( "plt.grid(True, linestyle='--', alpha=0.6)<BR>\n" );
document.write( "plt.gca().set_aspect('equal', adjustable='box')\r<BR>\n" );
document.write( "\n" );
document.write( "# Add legend<BR>\n" );
document.write( "plt.legend()\r<BR>\n" );
document.write( "\n" );
document.write( "# Save the plot<BR>\n" );
document.write( "plt.savefig('graph_3x_equals_y.png')\r<BR>\n" );
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document.write( "print(\"graph_3x_equals_y.png\")\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "```text?code_stdout&code_event_index=2<BR>\n" );
document.write( "graph_3x_equals_y.png\r<BR>\n" );
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document.write( "```\r<BR>\n" );
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document.write( "[image-tag: code-generated-image-0-1764864301049337471]\r<BR>\n" );
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document.write( "The equation $3x = y$ can be written in slope-intercept form as **$y = 3x + 0$**.\r<BR>\n" );
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document.write( "This means the line has:\r<BR>\n" );
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document.write( "  * A **y-intercept** of $0$, passing through the origin $(\mathbf{0, 0})$.<BR>\n" );
document.write( "  * A **slope** of $\mathbf{3}$ (or $\frac{3}{1}$), meaning for every $1$ unit you move to the right on the $x$-axis, you move $3$ units up on the $y$-axis.\r<BR>\n" );
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document.write( "The graph below shows the line $y = 3x$.\r<BR>\n" );
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document.write( "To plot this line, you can use the following points:\r<BR>\n" );
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document.write( "  * $(0, 0)$ (the y-intercept)<BR>\n" );
document.write( "  * $(1, 3)$ (using the slope: $x=0+1$, $y=0+3$)<BR>\n" );
document.write( "  * $(-1, -3)$ (using the slope: $x=0-1$, $y=0-3$) </SPAN>\n" );
document.write( "</TD></TR></TABLE>\n" );