Bài 2: Hệ hai phương trình bậc nhất hai ẩn
Hướng dẫn giải Bài 5 (Trang 11 SGK Toán Đại số 9, Tập 2)
<p>Đoán nhận số nghiệm của các hệ phương trình sau bằng hình học:</p>
<p>a)<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi mathvariant="normal">x</mi><mo>-</mo><mi mathvariant="normal">y</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi mathvariant="normal">x</mi><mo>-</mo><mn>2</mn><mi mathvariant="normal">y</mi><mo>=</mo><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math>; b)<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">y</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">y</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math></p>
<p><strong>Giải</strong></p>
<p>a) <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi>x</mi><mo>-</mo><mi>y</mi><mo>=</mo><mn>1</mn></mtd></mtr><mtr><mtd><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math></p>
<p>Vẽ <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>d</mi><mn>1</mn></msub></mfenced></math> : <math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mi>x</mi><mo>-</mo><mi mathvariant="normal">y</mi><mo>=</mo><mn>1</mn></math></p>
<p>Cho <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo> </mo><mo>⇒</mo><mi mathvariant="normal">y</mi><mo>=</mo><mo>-</mo><mn>1</mn></math>, ta được <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>A</mi><mo>(</mo><mn>0</mn><mo>;</mo><mo>-</mo><mn>1</mn><mo>)</mo></math>.</p>
<p>Cho <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></math>, ta được B <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>;</mo><mo> </mo><mn>0</mn></mrow></mfenced></math>.</p>
<p>Vì <math style="font-family:'Times New Roman'" xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>d</mi><mn>2</mn></msub></mfenced></math> : <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>-</mo><mn>2</mn><mi>y</mi><mo>=</mo><mo>-</mo><mn>1</mn></math></p>
<p>Cho <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>y</mi><mo>=</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><mo>,</mo><mo> </mo></math>ta được C <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>;</mo><mfrac><mn>1</mn><mn>2</mn></mfrac></mrow></mfenced></math>. </p>
<p>Cho <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mo>-</mo><mn>1</mn><mo>,</mo><mo> </mo></math>ta được D (<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>1</mn><mo>;</mo><mn>0</mn></mrow></mfenced></math></p>
<p>Hai đường thẳng cắt nhau tại điểm M có tọa độ <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mi>y</mi><mo>=</mo><mn>1</mn></mrow></mfenced></math></p>
<p>Thay <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo><mi>y</mi><mo>=</mo><mn>1</mn></math> vào các phương trình của hệ ta được:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>1</mn><mo>-</mo><mn>1</mn><mo>=</mo><mn>1</mn><mo> </mo><mfenced><mrow><mi>t</mi><mi>h</mi><mi>ỏ</mi><mi>a</mi></mrow></mfenced><mspace linebreak="newline"/><mn>1</mn><mo>-</mo><mn>2</mn><mo>.</mo><mn>1</mn><mo>=</mo><mo>-</mo><mn>1</mn><mfenced><mrow><mi>t</mi><mi>h</mi><mi>ỏ</mi><mi>a</mi></mrow></mfenced></math> </p>
<p>Vậy hệ phương trình có một nghiệm <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi mathvariant="normal">x</mi><mo>;</mo><mi mathvariant="normal">y</mi></mrow></mfenced><mo>=</mo><mfenced><mrow><mn>1</mn><mo>;</mo><mn>1</mn></mrow></mfenced></math>.</p>
<p><img class="wscnph" src="https://static.colearn.vn:8413/v1.0/upload/library/18022022/screenshot_1-eEnsow.png" /></p>
<p>b)<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced open="{" close=""><mtable columnalign="left"><mtr><mtd><mn>2</mn><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">y</mi><mo>=</mo><mn>4</mn></mtd></mtr><mtr><mtd><mo>-</mo><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">y</mi><mo>=</mo><mn>1</mn></mtd></mtr></mtable></mfenced></math></p>
<p>Vẽ <math style="font-family:'Times New Roman'" xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>d</mi><mn>1</mn></msub></mfenced><mo>:</mo><mo> </mo><mn>2</mn><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">y</mi><mo>=</mo><mn>4</mn></math></p>
<p>Cho <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>y</mi><mo>=</mo><mn>4</mn><mo>,</mo><mo> </mo></math>ta được A<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>;</mo><mn>4</mn></mrow></mfenced></math>.</p>
<p>Cho <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mn>2</mn></math>, ta được B <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>2</mn><mo>;</mo><mn>0</mn></mrow></mfenced></math></p>
<p>Vẽ <math style="font-family:'Times New Roman'" xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><msub><mi>d</mi><mn>2</mn></msub></mfenced></math>: <math xmlns="http://www.w3.org/1998/Math/MathML"><mo>-</mo><mi mathvariant="normal">x</mi><mo>+</mo><mi mathvariant="normal">y</mi><mo>=</mo><mn>1</mn></math></p>
<p>Cho <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>x</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>y</mi><mo>=</mo><mn>1</mn><mo>,</mo><mo> </mo></math>ta được C<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mn>0</mn><mo>;</mo><mn>1</mn></mrow></mfenced></math>.</p>
<p>Cho <math xmlns="http://www.w3.org/1998/Math/MathML"><mi>y</mi><mo>=</mo><mn>0</mn><mo>⇒</mo><mi>x</mi><mo>=</mo><mn>2</mn><mo>,</mo></math> ta được D<math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mo>-</mo><mn>1</mn><mo>;</mo><mn>0</mn></mrow></mfenced></math></p>
<p>Hai đường thẳng cắt nhau tại điểm N có tọa độ <math xmlns="http://www.w3.org/1998/Math/MathML"><mfenced><mrow><mi mathvariant="normal">x</mi><mo>=</mo><mn>1</mn><mo>;</mo><mo> </mo><mi mathvariant="normal">y</mi><mo>=</mo><mn>2</mn></mrow></mfenced></math></p>
<p>Thay <math xmlns="http://www.w3.org/1998/Math/MathML"><mtext>x=1, y=2</mtext></math> vào các phương trình của hệ ta được:</p>
<p><math xmlns="http://www.w3.org/1998/Math/MathML"><mn>2</mn><mo>.</mo><mn>1</mn><mo>+</mo><mn>2</mn><mo>=</mo><mn>4</mn><mo> </mo><mi>v</mi><mi>à</mi><mo> </mo><mo>-</mo><mn>1</mn><mo>+</mo><mn>2</mn><mo>=</mo><mn>1</mn><mo> </mo><mo>(</mo><mi>t</mi><mi>h</mi><mi>ỏ</mi><mi>a</mi><mo>)</mo></math></p>
<p><img class="wscnph" src="https://static.colearn.vn:8413/v1.0/upload/library/18022022/screenshot_2-k9wVXt.png" /></p>
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