Ôn tập chương III
Hướng dẫn giải Bài 41 (Trang 27 SGK Toán Đại số 9, Tập 2)
<p>Giải các hệ phương trình sau:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>a</mi><msqrt><mn>5</mn></msqrt><mo>-</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced><mi>y</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd><mfenced><mrow><mn>1</mn><mo>-</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced><mi>x</mi><mo>+</mo><mi>y</mi><msqrt><mn>5</mn></msqrt><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mo>;</mo><mo> </mo><mi>b</mi><mo>)</mo><mo> </mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mfrac><mrow><mn>2</mn><mi>x</mi></mrow><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mi>y</mi><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><msqrt><mn>2</mn></msqrt></mtd></mtr><mtr><mtd><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>+</mo><mfrac><mrow><mn>3</mn><mi>y</mi></mrow><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math></p>
<p><strong>Giải: </strong></p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>a</mi><mo>)</mo><mo> </mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>a</mi><msqrt><mn>5</mn></msqrt><mo>-</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced><mi>y</mi><mo>=</mo><mn>1</mn><mo> </mo><mfenced><mn>1</mn></mfenced></mtd></mtr><mtr><mtd><mfenced><mrow><mn>1</mn><mo>-</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced><mi>x</mi><mo>+</mo><mi>y</mi><msqrt><mn>5</mn></msqrt><mo>=</mo><mn>1</mn><mo> </mo><mfenced><mn>2</mn></mfenced></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mfenced><mn>2</mn></mfenced><mo>⇔</mo><mi>y</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mfenced><mrow><mn>1</mn><mo>-</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced><mi>x</mi></mrow><msqrt><mn>5</mn></msqrt></mfrac><mo> </mo><mfenced><mn>3</mn></mfenced></math></p>
<p>Thế (3) vào (1): </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><msqrt><mn>5</mn></msqrt><mo>-</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced><mfenced open="[" close="]"><mfrac><mrow><mn>1</mn><mo>-</mo><mfenced><mrow><mn>1</mn><mo>-</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced><mi>x</mi></mrow><msqrt><mn>5</mn></msqrt></mfrac></mfenced><mo>=</mo><mn>1</mn><mspace linebreak="newline"/><mo>⇔</mo><mn>5</mn><mi>x</mi><mo>-</mo><mfenced><mrow><mn>1</mn><mo>+</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced><mo>+</mo><mfenced><mrow><mn>1</mn><mo>-</mo><mn>3</mn></mrow></mfenced><mi>x</mi><mo>=</mo><msqrt><mn>5</mn></msqrt><mspace linebreak="newline"/><mo>⇔</mo><mn>3</mn><mi>x</mi><mo>=</mo><msqrt><mn>5</mn></msqrt><mo>+</mo><msqrt><mn>3</mn></msqrt><mo>+</mo><mn>1</mn><mspace linebreak="newline"/><mo>⇔</mo><mi>x</mi><mo>=</mo><mfrac><mrow><msqrt><mn>5</mn></msqrt><mo>+</mo><msqrt><mn>3</mn></msqrt><mo>+</mo><mn>1</mn></mrow><mn>3</mn></mfrac><mo>≈</mo><mn>1</mn><mo>,</mo><mn>66</mn></math></p>
<p>Thay vào (3) được: <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>-</mo><mfenced><mrow><mn>1</mn><mo>-</mo><msqrt><mn>3</mn></msqrt></mrow></mfenced><mstyle displaystyle="true"><mfrac><mrow><msqrt><mn>5</mn></msqrt><mo>+</mo><msqrt><mn>3</mn></msqrt><mo>+</mo><mn>1</mn></mrow><mn>3</mn></mfrac></mstyle></mrow><msqrt><mn>5</mn></msqrt></mfrac><mo>⇔</mo><mi>y</mi><mo>=</mo><mfrac><mrow><msqrt><mn>5</mn></msqrt><mo>+</mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>1</mn></mrow><mn>3</mn></mfrac></math></p>
<p>Vậy hệ có nghiệm <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>;</mo><mi>y</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mfrac><mrow><msqrt><mn>5</mn></msqrt><mo>+</mo><msqrt><mn>3</mn></msqrt><mo>+</mo><mn>1</mn></mrow><mn>3</mn></mfrac><mo>;</mo><mfrac><mrow><msqrt><mn>5</mn></msqrt><mo>+</mo><msqrt><mn>3</mn></msqrt><mo>-</mo><mn>1</mn></mrow><mn>3</mn></mfrac></mrow></mfenced></math></p>
<p>b) Đặt <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mi>u</mi></math> và <math xmlns="http://www.w3.org/1998/Math/MathML"><mfrac><mi>y</mi><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mi>v</mi></math>, ta có hệ phương trình</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi>u</mi><mo>+</mo><mi>v</mi><mo>=</mo><msqrt><mn>2</mn></msqrt></mtd></mtr><mtr><mtd><mi>u</mi><mo>+</mo><mn>3</mn><mi>v</mi><mo>=</mo><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced><mtable><mtr><mtd><mfenced><mn>1</mn></mfenced></mtd></mtr><mtr><mtd><mfenced><mn>2</mn></mfenced></mtd></mtr></mtable><mspace linebreak="newline"/><mfenced><mn>1</mn></mfenced><mo>⇔</mo><mi>v</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>2</mn><mi>u</mi><mo> </mo><mfenced><mn>3</mn></mfenced></math></p>
<p>Thay (3) vào (2): </p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>u</mi><mo>+</mo><mn>3</mn><mfenced><mrow><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>2</mn><mi>u</mi></mrow></mfenced><mo>=</mo><mo>-</mo><mn>1</mn><mspace linebreak="newline"/><mo>⇔</mo><mi>u</mi><mo>+</mo><mn>3</mn><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>6</mn><mi>u</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>⇔</mo><mn>5</mn><mi>u</mi><mo>=</mo><mn>3</mn><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>1</mn><mspace linebreak="newline"/><mo>⇔</mo><mi>u</mi><mo>=</mo><mfrac><mrow><mn>3</mn><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>1</mn></mrow><mn>5</mn></mfrac></math></p>
<p>Thay vào (3):</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mi>v</mi><mo>=</mo><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>2</mn><mo>.</mo><mfrac><mrow><mn>3</mn><msqrt><mn>2</mn></msqrt><mo>+</mo><mn>1</mn></mrow><mn>5</mn></mfrac><mo>⇔</mo><mi>v</mi><mo>=</mo><mfrac><mrow><mn>5</mn><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>6</mn><msqrt><mn>2</mn></msqrt><mo>-</mo><mn>2</mn></mrow><mn>5</mn></mfrac><mo>⇔</mo><mi>v</mi><mo>=</mo><mfrac><mrow><mo>-</mo><mn>2</mn><mo>-</mo><msqrt><mn>2</mn></msqrt></mrow><mn>5</mn></mfrac></math> nên hệ đã cho tương đương với:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mo> </mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mfrac><mi>x</mi><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow><mn>5</mn></mfrac></mtd></mtr><mtr><mtd><mfrac><mi>y</mi><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow></mfrac><mo>=</mo><mfrac><mrow><mo>-</mo><mn>2</mn><mo>-</mo><msqrt><mn>2</mn></msqrt></mrow><mn>5</mn></mfrac></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>5</mn><mi>x</mi><mo>=</mo><mfenced><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced><mfenced><mrow><mi>x</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mtd></mtr><mtr><mtd><mn>5</mn><mi>y</mi><mo>=</mo><mfenced><mrow><mo>-</mo><mn>2</mn><mo>-</mo><msqrt><mn>2</mn></msqrt></mrow></mfenced><mfenced><mrow><mi>y</mi><mo>+</mo><mn>1</mn></mrow></mfenced></mtd></mtr></mtable></mfenced><mspace linebreak="newline"/><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mfenced><mrow><mn>4</mn><mo>-</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow></mfenced><mi>x</mi><mo>=</mo><mn>1</mn><mo>+</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mtd></mtr><mtr><mtd><mfenced><mrow><mn>7</mn><mo>+</mo><msqrt><mn>2</mn></msqrt></mrow></mfenced><mi>y</mi><mo>=</mo><mo>-</mo><mn>2</mn><mo>-</mo><msqrt><mn>2</mn></msqrt></mtd></mtr></mtable></mfenced><mo>⇔</mo><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mi>x</mi><mo>=</mo><mfrac><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>4</mn><mo>-</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow></mfrac></mtd></mtr><mtr><mtd><mi>y</mi><mo>=</mo><mo>-</mo><mfrac><mrow><mn>2</mn><mo>+</mo><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>7</mn><mo>+</mo><msqrt><mn>2</mn></msqrt></mrow></mfrac></mtd></mtr></mtable></mfenced></math></p>
<p>Vậy hệ phương trình có nghiệm <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>;</mo><mi>y</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mfrac><mrow><mn>1</mn><mo>+</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>4</mn><mo>-</mo><mn>3</mn><msqrt><mn>2</mn></msqrt></mrow></mfrac><mo>;</mo><mo>-</mo><mfrac><mrow><mn>2</mn><mo>+</mo><msqrt><mn>2</mn></msqrt></mrow><mrow><mn>7</mn><mo>+</mo><msqrt><mn>2</mn></msqrt></mrow></mfrac></mrow></mfenced></math></p>
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