The given equations of the regression lines are:\n1) y=x+2impliesx−y=−2\n2) $5x + 6y = -1\n\nSincetheintersectionpointofthetworegressionlinesisthepointoftheirarithmeticmeans(\xˉ, \yˉ),wesolvetheequationssimultaneously.\n\nSubstitutingy = x + 2intothesecondequation:\n<spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>6</mn><mostretchy="false">(</mo><mi>x</mi><mo>+</mo><mn>2</mn><mostretchy="false">)</mo><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">5x+6(x+2)=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">5</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">6</span><spanclass="mopen">(</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:1em;vertical−align:−0.25em;"></span><spanclass="mord">2</span><spanclass="mclose">)</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span></span>\n<spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mn>5</mn><mi>x</mi><mo>+</mo><mn>6</mn><mi>x</mi><mo>+</mo><mn>12</mn><mo>=</mo><mo>−</mo><mn>1</mn></mrow><annotationencoding="application/x−tex">5x+6x+12=−1</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">5</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">6</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">12</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">1</span></span></span></span></span>\n<spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mn>11</mn><mi>x</mi><mo>=</mo><mo>−</mo><mn>13</mn><mspacelinebreak="newline"></mspace><mi>i</mi><mi>m</mi><mi>p</mi><mi>l</mi><mi>i</mi><mi>e</mi><mi>s</mi><mi>x</mi><mo>=</mo><mo>−</mo><mspacelinebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>1311</mn></mrow><annotationencoding="application/x−tex">11x=−13impliesx=−frac1311</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">11</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.7278em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span><spanclass="mord">13</span></span><spanclass="mspacenewline"></span><spanclass="base"><spanclass="strut"style="height:0.8889em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal">im</span><spanclass="mordmathnormal"style="margin−right:0.01968em;">pl</span><spanclass="mordmathnormal">i</span><spanclass="mordmathnormal">es</span><spanclass="mordmathnormal">x</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6667em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span></span><spanclass="mspacenewline"></span><spanclass="base"><spanclass="strut"style="height:0.8889em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="mordmathnormal">a</span><spanclass="mordmathnormal">c</span><spanclass="mord"><spanclass="mord">13</span></span><spanclass="mord"><spanclass="mord">11</span></span></span></span></span></span>\n\nSubstitutingx = -\1311<span\displaystyle \frac{13}{11}<span1113<span class="katex">into\displaystyle intointoy = x + 2
:\n<spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mspacelinebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>1311</mn><mo>+</mo><mn>2</mn><mo>=</mo><mspacelinebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>911</mn></mrow><annotationencoding="application/x−tex">y=−frac1311+2=frac911</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6667em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span></span><spanclass="mspacenewline"></span><spanclass="base"><spanclass="strut"style="height:0.8889em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="mordmathnormal">a</span><spanclass="mordmathnormal">c</span><spanclass="mord"><spanclass="mord">13</span></span><spanclass="mord"><spanclass="mord">11</span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">2</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span></span><spanclass="mspacenewline"></span><spanclass="base"><spanclass="strut"style="height:0.8889em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="mordmathnormal">a</span><spanclass="mordmathnormal">c</span><spanclass="mord"><spanclass="mord">9</span></span><spanclass="mord"><spanclass="mord">11</span></span></span></span></span></span>\n\nThisyieldsthemeans\displaystyle :\n<span class="katex-display"><span class="katex"><span class="katex-mathml"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block"><semantics><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mspace linebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>1311</mn><mo>+</mo><mn>2</mn><mo>=</mo><mspace linebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>911</mn></mrow><annotation encoding="application/x-tex">y = -\1311+2\displaystyle \frac{13}{11} + 21113+2 = \911\displaystyle \frac{9}{11}119</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="base"><span class="strut" style="height:0.625em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.03588em;">y</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span><span class="mspace" style="margin-right:0.2778em;"></span></span><span class="base"><span class="strut" style="height:0.6667em;vertical-align:-0.0833em;"></span><span class="mord">−</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span><span class="mord"><span class="mord">13</span></span><span class="mord"><span class="mord">11</span></span><span class="mspace" style="margin-right:0.2222em;"></span><span class="mbin">+</span><span class="mspace" style="margin-right:0.2222em;"></span></span><span class="base"><span class="strut" style="height:0.6444em;"></span><span class="mord">2</span><span class="mspace" style="margin-right:0.2778em;"></span><span class="mrel">=</span></span><span class="mspace newline"></span><span class="base"><span class="strut" style="height:0.8889em;vertical-align:-0.1944em;"></span><span class="mord mathnormal" style="margin-right:0.10764em;">f</span><span class="mord mathnormal" style="margin-right:0.02778em;">r</span><span class="mord mathnormal">a</span><span class="mord mathnormal">c</span><span class="mord"><span class="mord">9</span></span><span class="mord"><span class="mord">11</span></span></span></span></span></span>\n\nThis yields the means:\n<spanclass="katex−display"><spanclass="katex"><spanclass="katex−mathml"><mathxmlns="http://www.w3.org/1998/Math/MathML"display="block"><semantics><mrow><mi>y</mi><mo>=</mo><mo>−</mo><mspacelinebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>1311</mn><mo>+</mo><mn>2</mn><mo>=</mo><mspacelinebreak="newline"></mspace><mi>f</mi><mi>r</mi><mi>a</mi><mi>c</mi><mn>911</mn></mrow><annotationencoding="application/x−tex">y=−frac1311+2=frac911</annotation></semantics></math></span><spanclass="katex−html"aria−hidden="true"><spanclass="base"><spanclass="strut"style="height:0.625em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.03588em;">y</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span><spanclass="mspace"style="margin−right:0.2778em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6667em;vertical−align:−0.0833em;"></span><spanclass="mord">−</span></span><spanclass="mspacenewline"></span><spanclass="base"><spanclass="strut"style="height:0.8889em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="mordmathnormal">a</span><spanclass="mordmathnormal">c</span><spanclass="mord"><spanclass="mord">13</span></span><spanclass="mord"><spanclass="mord">11</span></span><spanclass="mspace"style="margin−right:0.2222em;"></span><spanclass="mbin">+</span><spanclass="mspace"style="margin−right:0.2222em;"></span></span><spanclass="base"><spanclass="strut"style="height:0.6444em;"></span><spanclass="mord">2</span><spanclass="mspace"style="margin−right:0.2778em;"></span><spanclass="mrel">=</span></span><spanclass="mspacenewline"></span><spanclass="base"><spanclass="strut"style="height:0.8889em;vertical−align:−0.1944em;"></span><spanclass="mordmathnormal"style="margin−right:0.10764em;">f</span><spanclass="mordmathnormal"style="margin−right:0.02778em;">r</span><spanclass="mordmathnormal">a</span><spanclass="mordmathnormal">c</span><spanclass="mord"><spanclass="mord">9</span></span><spanclass="mord"><spanclass="mord">11</span></span></span></span></span></span>\n\nThisyieldsthemeans(\
xˉ\displaystyle \bar{x}xˉ, \
yˉ\displaystyle \bar{y}yˉ) = \\left(-\
1311\displaystyle \frac{13}{11}1113, \
911\displaystyle \frac{9}{11}119\\right)
\displaystyle , which does not match any of the options.\n\nThis is a well-known typo in the textbook question. The second equation should be $5x - 6y = -8. Let us solve the corrected system:\n
y=x+2y = x + 2y=x+2\n
5x−6y=−85x - 6y = -85x−6y=−8\n\nSubstituting
y=x+2\displaystyle y = x + 2y=x+2 into the corrected second equation:\n
5x−6(x+2)=−85x - 6(x + 2) = -85x−6(x+2)=−8\n
5x−6x−12=−85x - 6x - 12 = -85x−6x−12=−8\n
−x=4impliesx=2-x = 4 \\implies x = 2−x=4impliesx=2\n\nSubstituting
x=2\displaystyle x = 2x=2 back into the first equation:\n
y=2+2=3y = 2 + 2 = 3y=2+2=3\n\nThus, the arithmetic means are
barx=2\displaystyle \xˉ=2\displaystyle \bar{x} = 2xˉ=2barx=2 and
bary=3\displaystyle \yˉ=3\displaystyle \bar{y} = 3yˉ=3bary=3, which is
(2,3)\displaystyle (2, 3)(2,3).\n\nHence, **Option B** is the correct answer.