document.write( "<A HREF=/cgi-bin/jump-to-question.mpl?question=1210484>Question 1210484</A>: <I><SPAN STYLE=\"word-break: break-all;\"> How would you graph y=-2/3+5 and -2x+4y=-8 with a point of (6,1)<BR>\n" );
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document.write( "</I><HR><TABLE width=100%><TR><TD><B><A HREF=http://www.algebra.com/>Algebra.Com's</A> Answer #852857 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=852857\">put this solution on YOUR website!</A><BR><SPAN STYLE=\"word-break: break-all;\"> It looks like you want to **graph two lines** and determine if the point $(6, 1)$ lies on their intersection.\r<BR>\n" );
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document.write( "The first equation, $y = -2/3 + 5$, is simply a horizontal line since it's a constant value. The second equation, $-2x + 4y = -8$, is a linear equation in standard form.\r<BR>\n" );
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document.write( "Let's first clarify and re-write the equations for graphing.\r<BR>\n" );
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document.write( "## 1. Analyze and Simplify the Equations\r<BR>\n" );
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document.write( "### Equation 1: $y = -2/3 + 5$\r<BR>\n" );
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document.write( "This equation simplifies to a single constant:<BR>\n" );
document.write( "$$y = -\frac{2}{3} + \frac{15}{3}$$<BR>\n" );
document.write( "$$y = \frac{13}{3} \approx 4.33$$<BR>\n" );
document.write( "This is a **horizontal line** where every point has a y-coordinate of $\frac{13}{3}$.\r<BR>\n" );
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document.write( "### Equation 2: $-2x + 4y = -8$\r<BR>\n" );
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document.write( "It's easiest to graph this line using the slope-intercept form ($y = mx + b$).\r<BR>\n" );
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document.write( "1.  **Isolate $y$:**<BR>\n" );
document.write( "    $$4y = 2x - 8$$<BR>\n" );
document.write( "2.  **Divide by 4:**<BR>\n" );
document.write( "    $$y = \frac{2}{4}x - \frac{8}{4}$$<BR>\n" );
document.write( "    $$y = \frac{1}{2}x - 2$$<BR>\n" );
document.write( "    This line has a **y-intercept ($b$) of $-2$** and a **slope ($m$) of $\frac{1}{2}$** (up 1, right 2).\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 2. Graphing the Lines\r<BR>\n" );
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document.write( "To graph these lines, you'll plot points and draw the lines on a coordinate plane.\r<BR>\n" );
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document.write( "### Line 1: $y = \frac{13}{3}$\r<BR>\n" );
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document.write( "1.  **Plot the y-intercept:** $(\mathbf{0, \frac{13}{3}})$ or $(0, 4.33)$.<BR>\n" );
document.write( "2.  **Draw a horizontal line** passing through that point.<BR>\n" );
document.write( "    * Example points: $(3, \frac{13}{3})$, $(-3, \frac{13}{3})$.\r<BR>\n" );
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document.write( "### Line 2: $y = \frac{1}{2}x - 2$\r<BR>\n" );
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document.write( "1.  **Plot the y-intercept:** $(\mathbf{0, -2})$.<BR>\n" );
document.write( "2.  **Use the slope** ($m = \frac{1}{2}$) to find other points:<BR>\n" );
document.write( "    * From $(0, -2)$, go up 1 and right 2 to find $(\mathbf{2, -1})$.<BR>\n" );
document.write( "    * From $(2, -1)$, go up 1 and right 2 to find $(\mathbf{4, 0})$.<BR>\n" );
document.write( "3.  **Draw a straight line** through these points.\r<BR>\n" );
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document.write( "---\r<BR>\n" );
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document.write( "## 3. Check the Given Point $(6, 1)$\r<BR>\n" );
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document.write( "Finally, let's see if the point $(6, 1)$ lies on **either** line.\r<BR>\n" );
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document.write( "### Check Line 1: $y = \frac{13}{3}$<BR>\n" );
document.write( "Substitute $y=1$:<BR>\n" );
document.write( "$$1 = \frac{13}{3}$$<BR>\n" );
document.write( "This is **False**. The point $(6, 1)$ is **not** on Line 1.\r<BR>\n" );
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document.write( "### Check Line 2: $y = \frac{1}{2}x - 2$<BR>\n" );
document.write( "Substitute $x=6$ and $y=1$:<BR>\n" );
document.write( "$$1 = \frac{1}{2}(6) - 2$$<BR>\n" );
document.write( "$$1 = 3 - 2$$<BR>\n" );
document.write( "$$1 = 1$$<BR>\n" );
document.write( "This is **True**. The point $(6, 1)$ **is** on Line 2.\r<BR>\n" );
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document.write( "The point $(6, 1)$ is only on the line $y = \frac{1}{2}x - 2$.\r<BR>\n" );
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document.write( "Would you like to find the exact point where these two lines intersect? </SPAN>\n" );
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